## Highlights - I first learned Serre’s definition of intersection multiplicity from Mel Hochster, back when I was an undergraduate\. I was immediately intrigued by this surprising connection between homological algebra and geometry (Daniel Dugger 3) - If you play around with some simple examples, an idea for defining intersection multiplicities comes up naturally\. It is i\(Z, W ; P \) = dim C h C [x1 , \. \. \. , xn ]/\(f1 , \. \. \. , fk , g1 , \. \. \. , gl \) i P \(1\.1\) i\(Z, W ; P \) = dimC C[x1 , \. \. \. , xn ]/\(f1 , \. \. \. , fk , g1 , \. \. \. , gl \) \. Here the subscript P indicates localization of the given ring at the maximal ideal \(x1 − p1 , \. \. \. , xn − pn \) where P = \(p1 , \. \. \. , pn \)\. (Daniel Dugger 5) - Example 1\.2\. Let f = y − x2 and g = y\. This is our example of the parabola and the tangent line at its vertex\. The point P = \(0, 0\) is the only intersection point, and our definition tells us to look at the ring C [x, y]/\(y − x 2 , y\) ∼ = ∼ = C[x]/\(x 2 \)\. As a vector space over C this is two-dimensional, with basis 1 and x\. So our definition gives i\(Z, W ; P \) = 2 as desired\. [Note that technically we should localize at the ideal \(x, y\), which corresponds to localization at \(x\) in C [x]/\(x 2 \); however, this ring is already local and so the localization has no effect] (Daniel Dugger 5) - Note the appearance of \(x − 1\) with multiplicity two in the above factorization\. The fact that we had a tangent line at x = 1 guaranteed that the multiplicity would be strictly larger than one (Daniel Dugger 6) - Serre discovered the correct formula for the interesection multiplicity [S]\. His formula is as follows\. If we set R = C [x1 , \. \. \. , xn ] then i\(Z, W ; P \) = ∞ X j=0 \(−1\) j dim h Tor R j R/\(f1 , \. \. \. , fk \), R/\(g1 , \. \. \. , g l \) i P \. (Daniel Dugger 7) - An algebraist who looks at \(1\.5\) will immediately notice some possible generalizations\. The R/\(f \) and R/\(g\) terms can be replaced by any finitely-generated module M and N , as long as the Torj \(M, N \) modules are finite-dimensional over C \. For this it turns out to be enough that M ⊗R N be finite-dimensional over C \. Also, we can replace C [x1 , \. \. \. , xn ] with any ring having the property that all finitely-generated modules have finite projective dimension—necessary so that the alternating sum of \(1\.5\) is finite\. Such rings are called regular\. (Daniel Dugger 7) - Let R be a regular, local ring \(all rings are assumed to be commutative and Noetherian unless otherwise noted\)\. Let M and N be finitely-generated modules over R such that M ⊗R N has finite length\. This implies that all the Torj \(M, N \) modules also have finite length\. Define e\(M, N \) = ∞ X j=0 \(−1\) j X \(−1\)j ` Torj \(M, N \) \(1\.6\) e\(M, N \) = and call this the intersection multiplicty of the modules M and N \. (Daniel Dugger 8) - Based on geometric intuition, Serre made the following conjectures about the above situation: \(1\) dim M + dim N ≤ dim R always \(2\) e\(M, N \) ≥ 0 always \(3\) If dim M + dim N < dim R then e\(M, N \) = 0\. \(4\) If dim M + dim N = dim R then e\(M, N \) > 0\. Serre proved all of these in the case that R contains a field, the so-called “geometric case” \(some non-geometric examples for R include power series rings over the padic integers Zp \)\. Serre also proved \(1\) in general\. Conjecture \(3\) was proven in the mid 80s by Roberts and Gillet-Soule \(independently\), using some sophisticated topological ideas that were imported into algebra\. Conjecture \(2\) was proven by Gabber in the mid 90s, using some high-tech algebraic geometry\. Conjecture \(4\) is still open\. (Daniel Dugger 8) - There are certain generalized cohomology theories—called complex-oriented — which have a close connection to geometry and intersection theory\. Any such cohomology can be used to detect intersection multiplicities\. (Daniel Dugger 8) - Topological K-theory is a complex-oriented cohomology theory\. Elements of the groups K ∗ \(X\) are specified by vector bundles on X, or more generally by bounded chain complexes of vector bundles on X\. Fundamental classes for complex submanifolds of X are given by resolutions\. (Daniel Dugger 8) - When X is an algebraic variety there is another version of K-theory called algebraic K-theory, which we might denote K ∗ alg \(X\)\. The analogs of vector bundles are locally free coherent sheaves, or just finitely-generated projective modules when X is affine\. Thus, in the affine case elements of K ∗ alg \(X\) can be specified by bounded chain complexes of finitely-generated projective modules\. This is the main connection between homological algebra and K-theory (Daniel Dugger 8) - This (Daniel Dugger 8) - \(4\) Serre’s definition of intersection multiplicities essentially comes from the intersection product in K-homology, which is the cup product in K-cohomology translated to homology via Poincaré Duality\. (Daniel Dugger 9) - Theorem 2\.1 \(Hilbert Syzygy Theorem\)\. Let k be a field and let R be k[x1 , \. \. \. , xn ] \(or any localization of this ring\)\. Then every finitely-generated R-module has a free resolution of length at most n\. (Daniel Dugger 10) - We mention it here because it implies that Torj \(M, N \) = 0 for j > n\. Therefore the sum in Serre’s formula is actually finite\. More generally, a ring is called regular if every finitely-generated module has a finite, projective resolution\. It is a theorem that localizations of regular rings are again regular\. Hilbert’s Syzygy Theorem simply says that polynomial rings over a field are regular\. (Daniel Dugger 10) - The importance of this observation is that it tells us that the Tor’s in Serre’s formula may all be taken over the ring RP \. So we might as well work over this ring from beginning to end (Daniel Dugger 10) - Lemma 2\.2\. Suppose that 0 → M 0 → M → M 00 → 0 is a short exact sequence of R-modules\. Then e\(M, N \) = e\(M 0 , N \)+e\(M 00 , N \), assuming all three multiplicities are defined \(that is, under the assumption that dim C \(M ⊗ N \) < ∞ and similarly with M replaced by M 0 and M 00 \)\. (Daniel Dugger 11) - Lemma 2\.2 is referred to as the additivity of intersection multiplicities\. Of course the additivity holds equally well in the second variable, by the same argument\. While exploring ideas in this general area, Grothendieck hit upon the idea of inventing a group that captures all the additive invariants of modules\. Any invariant such as e\(−, N \) would then factor through this group\. Here is the definition: Definition 2\.3\. Let R be any ring\. Let F\(R\) be the free abelian group with one generator [M ] for every isomorphism class of finitely-generated R-module M \. Let G\(R\) be the quotient of F\(R\) by the subgroup generated by all elements [M ] − [M 0 ] − [M 00 ] for every short exact sequence 0 → M 0 → M → M 00 → 0 of finitely-generated Rmodules\. The group G\(R\) is called the Grothendieck group of finitely-generated R-modules\. (Daniel Dugger 11) - 1\) Suppose R = F , a field\. Clearly G\(F \) is generated by [F ], since every finitelygenerated F -module has the form F n (Daniel Dugger 12) - More generally, suppose that R is a domain\. The rank of an R-module M is defined to be the dimension of M ⊗R QF \(R\) over QF \(R\), where QF \(R\) is the quotient field\. (Daniel Dugger 12) - Let G be a finite group, and let R = C [G] be the group algebra\. So R-modules are just representations of G on complex vector spaces (Daniel Dugger 12) - For a not-so-simple example, let R be the ring of integers in a number field\. It turns out that G\(R\) ∼ = ∼ = Z ⊕ Cl\(R\), where Cl\(R\) is the ideal class group of R\. This class group contains some sophisticated number-theoretic information about R\. It is known to always be torsion, and it is usually nontrivial\. (Daniel Dugger 12) - All finitely-generated R-modules have a finite composition series, and so we can take the Jordan-Hölder length; this is the same as `\(A\) = dim Z/p A/pA + dim Z/p pA\. With some trouble one can check that this is indeed an additive invariant \(or refer to the Jordan-Hölder theorem\), and of course `\( Z /p\) = 1 (Daniel Dugger 13) - Definition 2\.7\. Let R be any ring\. Let FK \(R\) be the free abelian group with one generator [P ] for every isomorphism class of finitely-generated, projective Rmodule M \. Let K\(R\) be the quotient of FK \(R\) by the subgroup generated by all elements [P ] − [P 0 ] − [P 00 ] for every short exact sequence 0 → P 0 → P → P 00 → 0 of finitely-generated projectives\. The group K\(R\) is called the Grothendieck group of finitely-generated projective modules\. Every short exact sequence of projectives is actually split, so we could also have defined K\(R\) by imposing the relations [P ⊕ Q] = [P ] + [Q] for every two finitelygenerated projectives P and Q\. This makes it a little easier to understand when two modules represent the same class in K\(R (Daniel Dugger 13) - Proposition 2\.8\. Let P and Q be finitely-generated projective R-modules\. Then [P ] = [Q] in K\(R\) if and only if there exists a finitely-generated projective module W such that P ⊕ W ∼ = ∼ = Q ⊕ W \. In fact, the same remains true if we require W to be free instead of projective\. (Daniel Dugger 14) - For the last statement in the proposition, just observe that since W is projective it is a direct summand of a free module\. That is, there exists a module W 0 such that W ⊕W 0 is finitely-genereated and free (Daniel Dugger 14) - Since projective modules are flat, the product [P ]·[Q] = [P ⊗R Q] is additive and so extends to a product K\(R\)⊗K\(R\) → K\(R\)\. (Daniel Dugger 14) - Remark 2\.9\. Given the motivation of having the tensor product give a ring structure, one might wonder why we used projective modules to define K\(R\) rather than flat modules\. We could have done so, but for finitely-generated modules over commutative, Noetherian rings, being flat and projective are equivalent notions (Daniel Dugger 14) - Theorem 2\.10\. If R is regular, then α : K\(R\) → G\(R\) is an isomorphism\. (Daniel Dugger 14) - Suppose Q• → M → 0 is another finite, projective resolution of M \. Use the Comparison Theorem of homological algebra to produce a map of chain complexes (Daniel Dugger 15) - Let T• be the mapping cone of f : P• → Q• \. Recall this means that Tj = Qj ⊕ Pj−1 , with the differential defined by dT \(a, b\) = dQ \(a\) + \(−1\) |b| f \(b\), dP \(b\) \. (Daniel Dugger 15) - There is a short exact sequence of chain complexes 0 → Q ,→ T → ΣP → 0 where ΣP denotes a copy of P in which everything has been shifted up a dimension \(so that \(ΣP \)n = Pn−1 \)\. (Daniel Dugger 15) - For any ring R, we have the group K\(R\) which also comes to us with an easily-defined ring structure ⊗\. We also have the group G\(R\)—but this does not have any evident ring structure\. When R is regular, there is an isomorphism K\(R\) → G\(R\) which allows one to transplant the ring structure from K\(R\) onto G\(R\): and this leads us directly to our alternating-sum-of-Tors (Daniel Dugger 16) - from K\(R\) onto G\(R\): and this leads us directly to our alternating-sum-of-Tors\. This situation is very reminiscent of something you have seen in a basic algebraic topology course\. When X is a \(compact, oriented\) manifold, there were early attempts to put a ring structure on H∗ \(X\) coming from the intersection product\. This is technically very difficult\. In modern times one avoids these technicalities by instead introducing the cohomology groups H ∗ \(X\), and here it is easy to define a ring structure: the cup product\. When X is a compact, oriented manifold one has the Poincaré Duality isomorphism H ∗ \(X\) → H∗ \(X\) given by capping with the fundamental class, and this lets one transplant the cup product onto H∗ \(X\)\. This is the modern approach to intersection theory\. (Daniel Dugger 16) - The parallels here are intriguing: K\(R\) is somehow like H ∗ \(X\), and G\(R\) is somehow like H∗ \(X\)\. The regularity condition is like being a manifold\. (Daniel Dugger 16) - [The reader might wonder what happened to the assumptions of compactness and orientability\. Neither of these is really needed for Poincaré Duality, as long as one does things correctly\. For the version of Poincaré Duality for noncompact manifolds one needs to replace ordinary homology with Borel-Moore homology—this is similar to singular homology, but chains are permitted to have infinitely many terms if they stretch out to infinity\. For non-orientable manifolds one needs to use twisted coefficients (Daniel Dugger 16) - out (Daniel Dugger 16) - Hilbert’s Nullstellensatz says that points of C n are in bijective correspondence with maximal ideals in R: the bijection sends q = \(q1 , \. \. \. , qn \) to mq = \(x1 −q1 , \. \. \. , xn −qn \)\. With a little work one can generalize this bijection\. If S ⊆ C n is any subset, define I\(S\) = {f ∈ R | f \(x\) = 0 for all x ∈ S}\. This is an ideal in R, in fact a radical ideal \(meaning that if f n ∈ I\(S\) then f ∈ I\(S\)\)\. In the other direction, if I ⊆ R is any ideal then define V \(I\) = {x ∈ C n | f \(x\) = 0 for all f ∈ I}\. Notice that V \(mq \) = {q} and I\({q}\) = mq \. (Daniel Dugger 17) - An algebraic set in C n is any subset of the form V \(I\) for some ideal I ⊆ R\. The algebraic sets form the closed sets for a topology on C n , called the Zariski topology\. One form of the Nullstellensatz says that V and I give a bijection between algebraic sets and radical ideals in R\. Under this bijection the prime ideals correspond to irreducible algebraic sets—ones that cannot be written as X ∪ Y where both X and Y are proper closed subsets\. Algebraic sets are also called algebraic subvarieties\. The above discussion is summarized in the following table: Geometry Algebra C n or An nC nC C [x1 , \. \. \. , xn ] = R Points \(q1 , \. \. \. , qn \) Maximal ideals \(x1 − q1 , \. \. \. , xn − qn \) Algebraic sets Radical ideals Irreducible algebraic sets Prime ideals (Daniel Dugger 17) - The ring R is best thought of as the set of maps of varieties A n → A 1 , with pointwise addition and multiplication\. If we restrict to some irreducible subvariety X = V \(P \) ⊆ A n instead, then the ring of functions X → A 1 is R/P \. This ring of functions is commonly called the coordinate ring of X\. Much of the dictionary between A n and R discussed above adapts verbatim to give a dictionary between X and its coordinate ring: Geometry Algebra X = V \(P \) C [x1 , \. \. \. , xn ]/P = R/P Points in X Maximal ideals in R/P Algebraic subsets V \(I\) ⊆ X Radical ideals in R/P Irreducible algebraic sets V \(Q\) ⊆ X Prime ideals in R/P \. Note that ideals in R/P correspond bijectively to ideals in R containing P , and likewise for prime \(respectively, radical\) ideals\. (Daniel Dugger 17) - We need one last observation\. Passing from A n to A n+1 corresponds algebraically to passing from R to R[t]\. If X = V \(P \) ⊆ A n is an irreducible algebraic set, then X × A 1 ⊆ A n+1 is V \(P [t]\) where P [t] ⊆ R[t]\. That is, the coordinate ring of X is R/P and the coordinate ring of X × A 1 is R[t]/P [t] = \(R/P \)[t]\. We supplement our earlier tables with the following line: Geometry Algebra X X × A 1 S S[t] We have defined G\(−\) and K\(−\) as functors taking rings as their inputs, but we could also think of them as taking varieties \(or schemes\) as their inputs (Daniel Dugger 17) - Theorem 2\.15\. If R is Noetherian, the Grothendieck group G\(R\) is generated by the set of elements [R/P ] where P ⊆R is prime\. Before proving this result let us comment on the significance\. When X is a topological space, the groups H∗ \(X\) have a geometric presentation in terms of “cycles” and “homologies”\. The cycles are, of course, generators for the group\. The definition of G\(R\) doesn’t look anything like this, but Theorem 2\.15 says that the group is indeed generated by classes that have the feeling of “algebraic cycles” on the variety Spec R (Daniel Dugger 18) - It is worth pointing out that in H∗ \(X\) the cycles are strictly separated by dimension—the dimensions i cycles are confined to the single group Hi \(X\)—whereas in G\(R\) the cycles of different dimensions are all inhabiting the same group\. This is one of the main differences between K-theory and singular homology/cohomology\. (Daniel Dugger 18) - Lemma 2\.16\. Let R be a Noetherian ring\. For any finitely-generated R-module M , there exists a prime ideal P ⊆R and an embedding R/P ,→ M \. Equivalently, there is some z∈M whose annihilator is prime\. (Daniel Dugger 18) - If M is an R-module, write M [t] for the R[t]-module M ⊗R R[t]\. The functor M→ 7 M [t] is exact, because R[t] is flat over R \(in fact, it is even free\)\. So we have an induced map α : G\(R\) → G\(R[t]\) given by [M ] 7→ [M [t]]\. Theorem 2\.18 \(Homotopy invariance\)\. If R is Noetherian, α : G\(R\) → G\(R[t]\) is an isomorphism\. We comment on the name “homotopy invariance” for the above result\. If X = Spec R then Spec R[t] = X × A 1 , so the result says that G\(−\) gives the same values on X and X × A 1 \. This is reminiscent of a functor on topological spaces giving the same values on X and X × I\. (Daniel Dugger 19) - Recall that a module is projective if and only if it is a direct summand of a free module\. So free modules are projective, and for almost all applications in homological algebra one can get by with using only free modules\. Consequently, it is common not to know many examples of non-free projectives\. We begin this section by remedying this\. (Daniel Dugger 20) - Let R = Z /6\. Since Z /2 ⊕ Z /3 ∼ = ∼ = Z/6, both Z/2 and Z/3 are projective R-modules—and √ they are clearly√ free\. not (Daniel Dugger 21) - This example generalizes: if D is a Dedekind domain \(such as the ring of integers in an algebraic number field\) then every ideal I ⊆ D is projective\. Non-principal ideals are never free\. (Daniel Dugger 21) - Let R = R [x, y, z]/\(x 2 + y 2 + z 2 − 1\)\. If C\(S 2 \) denotes the ring of continuous functions S 2 → R , note that we may regard R as sitting inside of C\(S 2 \): it is the subring of polynomial functions on the 2-sphere\. The connections with the topology of the 2-sphere will be important below\. Let π : R 3 → R be the map π\(f, g,h\) = xf + yg + zh\. That is, π is leftmultiplication by the matrix x y z \. Let T be the kernel of π: 0 → T ,→ R 3 π π −→ R → 0\. (Daniel Dugger 21) - The map π is split via χ : R → R3 sending 1 7→ \(x, y, z\)\. We conclude that T ⊕R ∼ = ∼ = R 3 , so T is projective\. (Daniel Dugger 21) - Note that T is, in some sense, an algebraic analog of the tangent bundle of S 2 \. These parallels between projective modules and vector bundles are very important, (Daniel Dugger 22) - This example is based on the Möbius bundle over S 1 \. Let S = R [x, y]/\(x 2 + y 2 − 1\) and let R ⊆ S be the span of the even degree monomials\. One should regard S as the ring of polynomial functions on the circle, and R is the ring of polynomial functions f \(x, y\) satisfying f \(x, y\) = f \(−x, −y\)\. So R is trying to be the ring of polynomial functions on R P 1 \(which happens to be homeomorphic to S 1 \)\. (Daniel Dugger 22) - A projective module P is called stably free if there exists a free module F such that P ⊕ F is free (Daniel Dugger 22) - \. It turns out that K\(R\) can be used to tell us whether such modules exist or not (Daniel Dugger 22) - Define the reduced Grothendieck group of R to be K e K\(R\) e = K\(R\)/h[R]i\. (Daniel Dugger 22) - In the 1950s, Serre conjectured that every finitely-generated projective over F [x1 , \. \. \. , xn ] is actually free\. As we will see later \(Remark 11\.5 below\), the motivation for this conjecture is inspired by topology and the connection between vector bundles and projective modules\. Quillen and Suslin independently proved Serre’s conjecture in the 1970s\. (Daniel Dugger 23) - Proposition 3\.1\. Let R be a commutative ring\. The following are equivalent: \(1\) K\(R\) ∼ = ∼ =Z \(2\) K e K\(R\) = 0 e \(3\) Every finitely-generated, projective R-module is stably-free\. (Daniel Dugger 23) - Remark 4\.3\. Theorem 4\.1 gives another parallel between G\(−\) and singular homology\. If X = Spec R then A = Spec R/f is a closed subscheme, and Spec f −1 R = X − A is the open complement (Daniel Dugger 24) - So the sequence in Theorem 4\.1 can be written as G\(A\) → G\(X\) → G\(X − A\) → 0\. This is somewhat reminiscent of the long exact sequence in singular homology · · · → H∗ \(A\) → H∗ \(X\) → H∗ \(X, A\) → · · · (Daniel Dugger 24) - The second thing is to recall something you probably learned in a basic algebra class, namely the Jordan-Hölder Theorem\. This says that given any two filtrations of M we can refine each one so that the two refinements have the same quotients up to reindexing\. (Daniel Dugger 28) - Definition 5\.2\. A map of chain complexes C• → D• is a quasi-isomorphism if the induced maps Hi \(C• \) → Hi \(D• \) are isomorphisms for all i ∈ Z\. Two chain complexes C• and D• are quasi-isomorphic, written C• ' D• , if there is a zig-zag of quasi-isomorphisms C• ∼ −→ J 1 ∼ ←− J 2 ∼ −→ · · · ∼ −→ J n ∼ ←− D• (Daniel Dugger 29) - Lemma 5\.3\. If P and Q are bounded below complexes of projectives, then every quasi-isomorphism is a chain homotopy equivalence\. (Daniel Dugger 29) - This lemma lets us replace the words “chain homotopy equivalence” with “quasiisomorphism” in any statement about bounded, projective complexes (Daniel Dugger 30) - \)\. The advantage of doing this is simply that quasi-isomorphisms are somewhat easier to identify than chain homotopy equivalences\. (Daniel Dugger 30) - why complicate things by making the defintion using complexes of arbitrary length? The answer comes from algebraic geometry\. Let X be a scheme and let U be an open subset of X\. Then the ‘correct’ way to define a relative K-theory group K\(X, U \) is to use bounded chain complexes of locally free sheaves on X that are exact on U \. When X = Spec R and U = Spec S −1 R then it happens that one can get the same groups using only complexes of length one—as we saw above\. But even for X = Spec R not every open subset is of this form\. A general subset will have the form U = \(Spec S −1 R\) ∪ \(Spec S −1 R\) ∪ · · · ∪ \(Spec S −1 R\), but to get the same relative K-group here one must use complexes of length at most d (Daniel Dugger 34) - When R is a regular ring all localizations S −1 R are also regular (Daniel Dugger 34) - Let R be a Noetherian ring, and let Z ⊆ Spec R be a Zariski closed set\. Recall that an R-module M is said to be supported on Z if MP = 0 for all primes P ∈ / Z\. One usually defines Supp M , the support of M , to be {P ∈ Spec R | MP 6= 0}\. (Daniel Dugger 34) - Let R be a discrete valuation ring \(a regular local ring whose maximal ideal is principal\), (Daniel Dugger 46) - Let D be a Dedekind domain—a regular ring of dimension one (Daniel Dugger 47) - In such a ring all nonzero primes are maximal ideals (Daniel Dugger 47) - The quotient of P 6=0 Z by the classes div\(x\) is called the divisor class group of D; it is isomorphic to the ideal class group from algebraic number theory\. Our short exact sequence shows that K e K e 0 \(D\) is also isomorphic to this group\. (Daniel Dugger 48) - Our next goal in these notes is to explore the idea of doing linear algebra locally over a fixed base space X\. To be slightly more precise, our objects of interest will be maps of spaces E → X where the fibers carry the structure of vector spaces; a map from E → X to E 0 → X is a continuous map F : E → E 0 , commuting with the maps down to X, such that F is a linear transformation on each fiber\. It turns out that much of linear algebra carries over easily to this enhanced setting\. But there are more isomorphism types of objects here, because the topology of X allows for some twisting in the vector space structure of the fibers\. The surprise is that studying these ‘twisted vector spaces’ over a base space X quickly leads to interesting homotopy invariants of X! Topological K-theory is a cohomology theory for topological spaces that arises out of this study of fibrewise linear algebra (Daniel Dugger 50) - Definition 8\.1\. A family of \(real\) vector spaces is a map p : E → X together with operations + : E ×X E → E and · : R × E → E making the two diagrams E ×X E HH H$ H H H H H H H E H + E ~ ~ ~ ~ ~ ~ ~ R × EF · F# F F F F F F F /E ~ ~ ~ ~ ~ ~ ~ ~ X X commute, and such that the operations make each fiber p −1 \(x\) into a real vector space over X\. One could write down the above definition completely category-theoretically, in terms of maps and commutative diagrams\. Essentially one is defining a “vector space object” in the category of spaces over X\. (Daniel Dugger 50) - notion is too wild to be of much use: there are too many ‘crazy’ families of vector spaces like this one\. One fixes this by adding a condition that forces the fibers to vary continuously, in a certain sense\. This is done as follows: Definition 8\.3\. A vector bundle is a family of vector spaces p : E → X such that for each x ∈ X there is a neighborhood x ∈ U ⊆ X, an n ∈ Z≥0 , and an isomorphism of families of vector spaces p −1 \(U \) F # F F F F F F F ∼ = ∼ = / U × R n w{ w w w w w w w w U Usually one simply says that a vector bundle is a family of vector spaces that is locally trivial\. The isomorphism in the above diagram is called a “local trivialization”\. (Daniel Dugger 51) - The family of vector spaces from Example 8\.2\(c\) perhaps makes it clear that this notion is too wild to be of much use: there are too many ‘crazy’ families of vector (Daniel Dugger 51) - Let X = R \. Let e1 , e2 be the standard basis for R 2 \. Let E ⊆ X × R 2 be the union of {\(x, re1 \) | x ∈ Q, r ∈ R} and {\(x, re2 \) | x ∈ X\Q, r ∈ R}\. Recall from \(a\) that X × R 2 → X is a family of vector spaces, and note that E becomes a sub-family of vector spaces under the same operations\. (Daniel Dugger 51) - Remark 8\.4\. Note that the n appearing in Definition 8\.3 depends on the point x\. It is called the rank of the vector bundle at x, and denoted rankx \(E\)\. It is easy to prove that the rank is constant on the connected components of X\. (Daniel Dugger 51) - Notation 8\.5\. If p : E → X is a family of vector spaces and A ,→ X is a subspace, then p −1 \(A\) → A is also a family of vector spaces\. We will usually write this restriction as E|A \. Note that if E is a vector bundle then so is E|A , by a simple argument\. The construction E|A is a special case of a pullback bundle, which we will discuss in Section 8\.9\. Of the families of vector spaces (Daniel Dugger 51) - we considered in Example 8\.2, only the trivial family from \(a\) is a vector bundle\. Before discussing more interesting examples, it will be useful to have a mechanism for deciding when a family of vector spaces is trivial\. If p : E → X is a family of vector spaces, a section of p is a map s : X → E such that ps = id\. The set of sections is denoted Γ\(E\), and this becomes a vector space using pointwise addition and multiplication in the fibers of E\. A collection of sections s1 , \. \. \. , sr is linearly independent if the vectors s1 \(x\), s2 \(x\), \. \. \. , sr \(x\) are linearly independent in Ex for every x ∈ X\. (Daniel Dugger 51) - Proposition 8\.6\. Let E → X be a family of vector spaces of constant rank n\. Then the family is trivial if and only if there is a linearly independent collection of sections s1 , s2 , \. \. \. , sn \. (Daniel Dugger 52) - \(a\) Let φ : R n → R n be a vector space isomorphism\. Let E 0 = [0, 1] × R n and let E be the quotient of E 0 by the relation \(0, v\) ∼ \(1, φ\(v\)\)\. Identifying S 1 with the quotient of [0, 1] by 0 ∼ 1, we obtain a map E → S 1 that is clearly a family of vector spaces\. We claim this is a vector bundle\. If x ∈ \(0, 1\) then it is evident that E is locally trivial at x, so the only point of concern is x = 0 = 1 ∈ S 1 \. Let e1 , \. \. \. , en be the standard basis for R n , and let si : [0, 41 41 ] → E 0 be the constant section whose value is ei \. Likewise, let s0 i i : \( 4 , 1] → E 0 be the constant section whose value is φ\(ei \)\. Projecting into E we obtain si \(0\) = s 0i \(1\), and so the sections si and s0 i patch together to give a section Si : U → E, where U = [0, 41 1 \) ∪ \( 34 , 1]\. The sections S1 , \. \. \. , Sn are independent and therefore give a local trivialization of E over U \. When n = 1 and φ\(x\) = −x the resulting bundle is the Möbius bundle M , depicted below: (Daniel Dugger 52) - Let X = R P n , and let L ⊆ X × R n+1 be the set L = {\(l, v\) | l ∈ R P n , x ∈ l}\. Then L is a subfamily of the trivial family, and we claim that it is a line bundle over X\. To see this, for any l ∈ X we must produce a local trivialization\. By symmetry it suffices to do this when l = he1 i\. Let U ⊆ RP n be the set of lines whose orthogonal projection to he1 i is nonzero\. Such a line contains a unique vector of the form e1 + u where e1 · u = 0\. Define s : U → L by sending l to \(l, e1 + u\) where e1 + u is the unique point on l described above\. This section is clearly nonzero everywhere, so it gives a trivialization of L|U \. Thus, we have proven that L is locally trivial and hence a vector bundle\. The bundle L is called the tautological line bundle over R P n \. Do not confuse this with the canonical line bundle over R P n that we will define shortly \(they are duals of each other\)\. Note that when n = 1 the bundle L is isomorphic to the Möbius bundle on S 1 \. (Daniel Dugger 52) - Projection to the first coordinate π : η → Grk \(V \) makes η into a rank k vector bundle, called the tautological bundle over Grk \(V \)\. (Daniel Dugger 53) - Pullback bundles can be slightly non-intuitive\. Let M → S 1 be the Möbius bundle, and let f : S 1 → S 1 be the map z 7→ z 2 \. We claim that f ∗ M ∼ = 1\. This is easiest to see if one uses the following model for M : M = n e iθ , re i θ θ 2 θ ∈ [0, 2π], r ∈ R o (Daniel Dugger 53) - We may form a new bundle E ⊕ E 0 , whose underlying topological space is just the pullback E ×X E 0 \. So a point in E ⊕ F is a pair \(e, e0 \) where p\(e\) = p0 \(e0 \)\. The rules for vector addition and scalar multiplication are the evident ones\. Note that the fiber of E ⊕ E 0 over a point x is simply Ex ⊕ Ex0 \. (Daniel Dugger 54) - More generally, any canonical construction one can apply to vector spaces may be extended to apply to vector bundles\. So one can talk about the bundles E ⊗ E 0 , the dual bundle E ∗ , the hom-bundle Hom\(E, E 0 \), the exterior product bundle /\ i E, and so on (Daniel Dugger 54) - Recall that if fA : A → Y and fB : B → Y are continuous maps that agree on A ∩ B then we may patch these together to get a continuous map f : X → Y provided that either \(i\) A and B are both closed, or \(ii\) A and B are both open\. This is a basic fact about topological spaces\. (Daniel Dugger 55) - closed, or \(ii\) A and B are both open\. This is a basic fact about topological spaces\. The analogous facts for vector bundles are very similar in the case of an open cover, but more subtle for closed covers\. Proposition 8\.15\. Let E → X be a family of vector spaces\. \(a\) If {Uα } is an open cover of X and each E|Uα is a vector bundle, then E is a vector bundle\. \(b\) Suppose {A, B} is a cover of X by closed subspaces, and that for every x ∈ A∩B and every open neighborhood x ∈ U ⊆ X there exists a neighborhood x ∈ V ⊆ U such that V ∩ A ∩ B ,→ V ∩ B has a retraction\. Then if E|A and E|B are both vector bundles, so is E\. (Daniel Dugger 55) - \(ii\) The two isomorphisms φγ,α and φγ,β ◦ φβ,α agree on their common domain of definition, which is Eα |Uα ∩Uβ ∩Uγ \. (Daniel Dugger 56) - Condition \(ii\) above is usually called the cocycle condition\. (Daniel Dugger 56) - To see why, consider the case where all of the Eα ’s are trivial bundles of rank n\. Then the data in the φα,β maps is really just the data of a map gα,β : Uα ∩ Uβ → GLn \(R\)\. These (Daniel Dugger 56) - gα,β maps are called transition functions\. (Daniel Dugger 57) - Condition \(ii\) is the requirement that the transition functions assemble to give a C Č Čech 1-cocycle with values in the group GLn \(R\) (Daniel Dugger 57) - We will see in a moment \(Corollary 8\.23\) that for real vector bundles over paracompact Hausdorff spaces one always has E ∼ = ∼ = E ∗ , although the isomorphism is not canonical\. This is not true for complex or quaternionic bundles, however (Daniel Dugger 57) - Let L → C P n be the tautological complex line bundle over C P n \. Its \(complex\) dual L ∗ is called the canonical line bundle over C P n \. (Daniel Dugger 57) - Whereas from a topological standpoint neither L nor L ∗ holds a preferential position over the other, in algebraic geometry there is an important difference between the two\. The difference comes from the fact that L ∗ has certain “naturally defined” sections, whereas L does not (Daniel Dugger 57) - For a point z = [z0 : · · · : zn ] ∈ CP n , Lz is the complex line in C n+1 spanned by \(z0 , \. \. \. , zn \)\. Given only z ∈ CP n there is no evident way of writing down a point on Lz , without making some kind of arbitrary choice; said differently, the bundle L does not have any easily-described sections\. In contrast, it is much easier to write down a functional on Lz \. For example, let φi be the unique functional on Lz that sends the point \(z0 , \. \. \. , zn \) to zi \. Notice that this description depends only on z ∈ C P n , not the point \(z0 , \. \. \. , zn \) ∈ Cn+1 that represents it; that is, the functional sending \(λz0 , \. \. \. , λzn \) to λzi is the same as φi \. In this way we obtain an entire C n+1 ’s worth of sections for L ∗ , by taking linear combinations of the φi ’s\. (Daniel Dugger 57) - To be clear, it is important to realize that L has plenty of sections—it is just that one cannot describe them by simple formulas\. The slogan to remember is that the bundle L ∗ has algebraic sections, whereas L does not\. In algebraic geometry the bundle L ∗ is usually denoted O\(1\), whereas L is denoted O\(−1\)\. More generally, O\(n\) denotes \(L∗ \)⊗n when n ≥ 0 \(so that O\(0\) is the trivial line bundle\), and denotes L ⊗\(−n\) when n < 0 (Daniel Dugger 57) - Definition 8\.21\. Let E → X be a real vector bundle\. An inner product on E is a map of vector bundles E ⊗ E → 1 that induces a positive-definite, symmetric, bilinear form on each fiber Ex \. A vector bundle with an inner product is usually called an orthogonal vector bundle\. (Daniel Dugger 58) - Every complex vector space may be equipped with a nondegenerate, symmetric bilinear form (Daniel Dugger 58) - Inner products on R n are in bijective correspondence with symmetric, positive-definite matrices A ∈ Mn×n \(R\), (Daniel Dugger 59) - by sending an inner product h−, −i to the matrix aij = hei , ej i (Daniel Dugger 59) - Now consider the fibration sequence On ,→ GLn \(R\) → GLn \(R\)/On \. The projection map sends a matrix P to P In P T = P P T \. The inclusion On ,→ GLn \(R\) is a homotopy equivalence by Gram-Schmidt, and so GLn \(R\)/On is weakly contractible\. Standard techniques show that this homogeneous space may be given the structure of a CW-complex (Daniel Dugger 59) - Suppose that E → X is a rank n real vector bundle with an inner product\. Choose a trivializing open cover {Uα }, and for each α fix an inner-productpreserving trivialization fα : E|Uα → Uα × Rn where the codomain has the standard inner product \(this is possible by Proposition 8\.25\)\. The transition functions gα,β : Uα ∩ Uβ → GLn \(R\) therefore factor through On , as they must preserve the inner product\. This process is usually referred to as reduction of the structure group\. (Daniel Dugger 59) - We claim that the projection map p1 : Z → W is a fiber bundle with fiber Rk\(n−k\) , but defer the proof for just a moment\. The fact that the fiber is contractible then shows that p1 is weak homotopy equivalence (Daniel Dugger 61) - Proposition 9\.2\. Let X be a paracompact space\. Then any surjection of bundles E F has a splitting\. (Daniel Dugger 61) - Proof\. Briefly, we choose local splittings and then use a partition of unity to patch them together (Daniel Dugger 61) - Proposition 9\.3\. Let X be any space, and let f : E → F be a map of vector bundles over X\. If f has constant rank then ker f , coker f , and im f are vector bundles\. (Daniel Dugger 62) - Corollary 9\.4\. Let X be a paracompact space\. Then any injection of bundles E ,→ F has a splitting\. (Daniel Dugger 62) - Proposition 9\.5\. Suppose that X is compact and Hausdorff\. Then every bundle is a subbundle of some trivial bundle\. (Daniel Dugger 62) - Lemma 9\.7\. Let E α α −→ F β −→ G be an exact sequence of vector bundles\. Then im α \(which equals ker β\) is a vector bundle\. (Daniel Dugger 63) - In this section we explore our first connection between topology and algebra\. We will see that vector bundles are closely related to projective modules\. (Daniel Dugger 64) - The assignment E 7→ Γ\(E\) gives a functor from vector bundles to C\(X\)-modules\. (Daniel Dugger 64) - When X is a space let C\(X\) denote the ring of continuous functions from X to R , where the addition and multiplication are pointwise (Daniel Dugger 64) - It is easy to check that Γ is a left-exact functor (Daniel Dugger 64) - If E → X is a vector bundle then of course the modules of the form Γ\(E\) are not just arbitrary C\(X\)-modules; there is something special about them\. It is easiest to say what this is under some assumptions on X: Proposition 10\.1\. If X is compact and Hausdorff, and E is a vector bundle over X, then Γ\(E\) if a finitely-generated, projective module over C\(X\)\. (Daniel Dugger 64) - That is, Γ\(E\) is a direct summand of a free module; hence it is projective\. (Daniel Dugger 64) - For the rest of this section we will assume that our base spaces are compact and Hausdorff\. Let hhVect\(X\)ii denote the category of vector bundles over X, and let hhMod −C\(X\)ii denote the category of modules over the ring C\(X\)\. (Daniel Dugger 64) - hhProj −C\(X\)ii denote the full subcategory of finitely-generated, projective modules\. Then Γ is a functor hhVect\(X\)ii → hhProj −C\(X\)ii\. It is proven in [Sw] that this is actually an equivalence: Theorem 10\.2 \(Swan’s Theorem\)\. Let X be a compact, Hausdorff space\. Then Γ : hhVect\(X\)ii → hhProj −C\(X\)ii is an equivalence of categories\. (Daniel Dugger 65) - To prove this result we need to verify two things: • Every finitely-generated projective over C\(X\) is isomorphic to Γ\(E\) for some vector bundle E\. • For every two vector bundles E and F , the induced map Γ : HomVect\(X\) \(E, F \) → HomC\(X\) \(ΓE, ΓF \) is a bijection\. (Daniel Dugger 65) - That is to say, we need to prove that Γ is surjective on isomorphism classes, and is fully faithful (Daniel Dugger 65) - Note that we have the evaluation map evx : Γ\(E\) → Ex \. This map clearly sends the submodule mx Γ\(E\) to zero\. Lemma 10\.4\. Assume that X is paracompact Hausdorff\. Then for any vector bundle E → X and any x ∈ X, the map evx : Γ\(E\)/mx Γ\(E\) ∼ = ∼ = = −→ Ex is an isomorphism\. (Daniel Dugger 66) - Our final goal is to prove that Γ is fully faithful\. To do this, it is useful to relate the fibers Ex of our bundle to an algebraic construction based on the module Γ\(E\)\. For each x ∈ X consider the evaluation map evx : C\(X\) → R, and let mx be the kernel\. The ideal mx ⊆ C\(X\) is maximal, since the quotient is a field\. (Daniel Dugger 66) - Proposition 10\.5\. Assume that X is paracompact Hausdorff\. Then for any vector bundles E and F over X, the map Γ : HomVect\(X\) \(E, F \) → HomC\(X\) \(ΓE, ΓF \) is a bijection\. (Daniel Dugger 66) - For a fixed n, let Vectn \(X\) denote the set of isomorphism classes of vector bundles on X\. It turns out that when X is a finite complex this set is always countable, and often finite\. It actually gives a homotopy invariant of the space X\. (Daniel Dugger 67) - Example 11\.3\. To give an idea how we will apply these results, let us think about vector bundles on S 1 \. Divide S 1 into an upper hemisphere D+ and a lower hemisphere D− , intersecting in two points\. Each of D+ and D− are contractible, so any vector bundle will be trivializable when restricted to these subspaces\. Given two elements α, β ∈ GLn \(R\), let En \(α, β\) be the vector bundle on S 1 obtained by taking n D+ and n D− and gluing them together via α and β at the two points on the equator\. The considerations of the previous paragraph tell us that every vector bundle on S 1 is of this form (Daniel Dugger 68) - In \(2\) we have used the fact that π0 \(GLn \(R\)\) = Z/2, with the isomorphism being given by the sign of the determinant\. (Daniel Dugger 68) - To summarize, from \(1\) and \(2\) it follows that isomorphism types for rank n bundles over S 1 are in bijective correspondence with the path components of GLn \(R\)\. (Daniel Dugger 69) - The methods of the above example apply in much greater generality, and with little change allow one to get control over vector bundles on any suspension\. (Daniel Dugger 69) - Remark 11\.5\. We have seen that all bundles on contractible spaces are trivial, and that there is a close connection between vector bundles and projective modules\. Recall that when k is a field then k[x1 , \. \. \. , xn ] is the algebraic analog of affine space A n , and that projectives over this ring correspond to algebraic vector bundles\. The analogy with topology is what led Serre to conjecture that all finitely-generated projectives over k[x1 , \. \. \. , xn ] are free (Daniel Dugger 70) - We have proven that if E is a vector bundle on X × I then i∗0 \(E\) ∼ = ∼ i = ∗ 1 \(E\)\. It is natural to wonder if this result has a converse, but stating such a thing is somewhat tricky (Daniel Dugger 70) - So we find ourselves in somewhat of a muddle\. Perhaps there is an interesting question here, but we don’t quite know how to ask it\. One approach is to restrict to a class of bundles where “equality” is something we can better control (Daniel Dugger 70) - We may view a vector bundle as a family of vector spaces indexed by the base space\. In general, we may view a map X → Y as a family of blah if each fiber is a blah\. (Daniel Dugger 71) - We naively hope that families of some mathematical object over X are in bijection with maps from X to some space, called the moduli space corresponding to that mathematical object\. (Daniel Dugger 71) - some space, called the moduli space corresponding to that mathematical object\. With this naive idea, we would hope that families over ∗ are in bijective correspondence with points of our moduli space\. However, this does not work since the moduli space of R n ’s is ∗\. (Daniel Dugger 71) - V ⊆ W then we get an induced inclusion of Grassmannians Grk \(V \) ,→ Grk \(W \)\. Consider the standard chain of inclusions of Euclidean spaces R k , and define the infinite Grassmannian Grn \(R ∞ \) to be the colimit of the induced sequence of finite Grassmannians: Grn \(R ∞ \) = colim k→∞ [Grn \(R k \)]\. (Daniel Dugger 71) - Define γn → Grn \(R∞ \) by γn = {\(V, x\) | V ⊂ R∞ , dim\(V \) = n, x ∈ V }\. This is the tautological vector bundle on the infinite grassmanian\. (Daniel Dugger 71) - To any map f : X → Grn \(R∞ \) we associate the pullback bundle f ∗ γn / γn X f / Grn \(R ∞ \)\. The assignment f 7→ f ∗ γn gives a map Hom\(X, Grn \(R∞ \)\) → Vectn \(X\)\. Observe that if f, g : X → Grn \(R∞ \) are homotopic maps, then f ∗ γn ∼ = g ∗ γn by Corollary 11\.2\(a\)\. In this way we have constructed a map φ : [X, Grn \(R ∞ \)] → Vectn \(X\)\. We will show that this is an isomorphism when X is compact and Hausdorff (Daniel Dugger 71) - Theorem 11\.8\. The map φ : [X, Grn \(R ∞ \)] → Vectn \(X\) is always injective, and is bijective when X is compact and Hausdorff\. (Daniel Dugger 71) - Proposition 11\.10\. Let X be a finite-dimensional CW-complex\. For real vector bundles, Vectn \(X\) → Vectn+1 \(X\) is a bijection for n ≥ dim X + 1 and a surjection for n = dim X\. For complex bundles, VectC n n C \(X\) → Vectn n+1 \(X\) is a bijection for n ≥ 21 dim X and a surjection for n ≥ 12 \(dim X − 1\)\. (Daniel Dugger 72) - Fix a space X\. If E → X is a vector bundle of rank n, then of course E ⊕ 1 is a vector bundle of rank n + 1\. We get a sequence of maps Vect0 \(X\) ⊕1 −→ Vect1 \(X\) ⊕1 −→ Vect2 \(X\) ⊕1 −→ · · · Are these maps injective? Surjective? Are there more and more isomorphism classes of vector bundles as one goes up in rank, or is it the case that all “large” rank vector bundles actually come from smaller ones via addition of a trivial bundle? (Daniel Dugger 72) - rank vector bundles actually come from smaller ones via addition of a trivial bundle? A homotopical analysis of classifying spaces allow us give some partial answers here\. (Daniel Dugger 72) - this section we explore the set of isomorphism classes Vectn \(S k \) for various values of k and n\. There are two important points\. First, for a fixed k these sets stablize for n 0\. Secondly, Bott was able to compute these stable values completely and found an 8-fold periodicity \(with respect to k\) in the case of real vector bundles, and a 2-fold periodicity in the case of complex bundles\. Bott’s periodicity theorems are of paramount importance in modern algebraic topology (Daniel Dugger 72) - In (Daniel Dugger 72) - 12\.1\. The clutching construction\. Let X be a pointed space, and let C+ and C− be the positive and negative cones in ΣX\. Fix n ≥ 0\. For a map f : X → GLn \(R\), let En \(f \) be the vector bundle obtained by gluing n|C+ and n|C− via the map f \(we use Corollary 8\.17\(b\) here\)\. Precisely, if x ∈ X and v belongs to the fiber of n C+ over x then we glue v to f \(x\) · v in the fiber of n C− over x\. This procedure for constructing vector bundles on ΣX is called clutching, and every bundle on ΣX arises in this way (Daniel Dugger 73) - Let us apply the above result when X is a sphere S k−1 \. We obtain a bijection Vectn \(S k \) ' πk−1 GLn \(R\)\. (Daniel Dugger 73) - Recall that On ,→ GLn \(R\) is a deformation retraction, as a consequence of the Gram-Schmidt process (Daniel Dugger 73) - When k > 2 any based map S k−1 → On must actually factor through the connected component of the identity, which is SOn \. So we have Vectn \(S k \) ∼ = ∼ = πk−1 GLn \(R\) = ∼ = πk−1 On ∼ ∼ = ∼ = πk−1 SOn (Daniel Dugger 73) - 12\.3\. Vector bundles on S 1 \. For k = 1 and n > 0 we have that Vectn \(S 1 \) ∼ = ∼ = π0 GLn \(R\) = Z/2, and we have previously seen in Example 11\.3 that the two isomorphism classes are represented by n and M ⊕ \(n − 1\) where M is the Möbius bundle\. (Daniel Dugger 73) - 12\.4\. Vector bundles on S 2 \. Here we have Vectn \(S 2 \) ∼ = ∼ = π1 SOn \. Recall that SO2 ∼ = ∼ = S 1 , and so we get Vect2 \(S 2 \) ∼ = ∼ = Z\. We claim that for n > 2 one has π1 SOn ∼ = ∼ = Z/2, so that we have the following: (Daniel Dugger 73) - Proposition 12\.5\. Vectn \(S 2 \) ∼ = ∼ = π1 \(SOn \) (Daniel Dugger 73) - For n = 3 recall that SO3 ∼ = ∼ = RP 3 , so that π1 \(SO3 \) ∼ = ∼ = Z/2\. To see the homeomorphism use the model R P 3 ∼ = ∼ = D 3 /∼ where the equivalence relation has x ∼ −x for x ∈ ∂D3 \. (Daniel Dugger 73) - Proof\. First of all SO1 = {1} and SO2 ∼ = ∼ = S 1 , so this takes care of n ≤ 2\. For n = 3 recall that SO3 ∼ = ∼ = RP 3 , so (Daniel Dugger 73) - Map D 3 → SO3 by sending a vector v to the rotation of R3 with axis hvi, through |v| · π radians, in the direction given by a right-hand-rule with the thumb pointed along v\. Note that this makes sense even for v = 0, since the corresponding rotation is through 0 radians\. For x ∈ ∂D3 this map sends x and −x to the same rotation, and so induces a map R P 3 → SO3 \. This is clearly a continuous bijection, and therefore a homeomorphism since the spaces are compact and Hausdorff\. (Daniel Dugger 74) - For n ≥ 4 one can use the long exact sequence associated to the fibration SOn−1 ,→ SOn S n−1 to deduce that π1 \(SOn \) ∼ = π1 \(SOn−1 \)\. (Daniel Dugger 74) - Definition 12\.6\. Let O\(n\) ∈ Vect2 \(S 2 \) be the vector bundle Efn where fn : S 1 → SO2 is a map of degree n\. Note that O\(0\) ∼ = 2\. The bundles O\(n\), n ∈ Z , give a complete list of the rank 2 bundles on S 2 (Daniel Dugger 74) - Putting all of this information together, the following table shows all the vector bundles on S 2 : n 1 2 3 4 5 6 Vectn \(S 2 \) 1 O\(n\), n ∈ Z 3, O\(1\) ⊕ 1 4, O\(1\) ⊕ 2 5, O\(1\) ⊕ 3 ··· The operation \(−\) ⊕ 1 moves us from one column of the table to the next (Daniel Dugger 74) - answer, namely what happens when one adds two rank 2 bundles \(all other sums can be figured out once one knows how to do these\): Theorem 12\.7\. O\(j\) ⊕ O\(k\) (Daniel Dugger 74) - It is a standard fact in topology that the group structure on [S 1 , SO4 ]∗ given by (Daniel Dugger 74) - pointwise multiplication agrees with the group structure given by concatenation of loops \(this is true with SO4 replaced by any topological group\)\. (Daniel Dugger 75) - 12\.8\. Vector bundles on S 3 \. Now we have to calculate π2 SOn \. This is trivial for n ≤ 2 \(easy\), and for n = 3 it also trivial: use SO3 ∼ = ∼ = RP 3 and the fibration sequence Z /2 ,→ S 3 R P 3 \. Finally, the fibration sequences SOn−1 ,→ SOn S n−1 now show that π2 SOn = 0 for all n\. We have proven Proposition 12\.9\. Vectn \(S 3 \) ∼ = ∼ = π2 \(SOn \) ∼ = ∼ = 0\. That is, every vector bundle on S 3 is trivializable\. (Daniel Dugger 75) - 12\.10\. Vector bundles on S 4 \. Once again, we are reduced to calculating π3 SOn \. Eventually one expects to get stuck here, but so far we have been getting lucky so let’s keep trying (Daniel Dugger 75) - 12\.12\. Vector bundles on S k \. Although we can not readily do the calculations for k > 4, at this point one sees the general pattern\. One must calculate πk−1 SOn for each n, and these groups vary for a while but eventually stabilize\. In fact, πi SOn ∼ = ∼ = πi SOn+1 for i + 1 < n\. The calculation of these stable groups was an important problem back in the 1950s, that was eventually solved by Bott\. (Daniel Dugger 75) - The colimit of this sequence is denoted O and called the stable orthogonal group\. The homotopy groups of O are the stable values that we encountered above\. We computed the first few: π0 O = Z/2, π1 O = Z /2, π2 O = 0\. And we stated, without proof, that π3 O = Z\. Bott’s calculation showed the following: (Daniel Dugger 76) - The pattern is 8-fold periodic: πi+8 O ∼ = ∼ = πi O for all i ≥ 0\. One is supposed to remember the pattern of groups to the tune of “Twinkle, Twinkle, Little Star”: zee two zee two ze ro zee ze ro ze ro ze ro zee\. (Daniel Dugger 76) - 12\.13\. Complex vector bundles on spheres\. One can repeat the above analysis for complex vector bundles on a sphere\. One finds that VectC n n \(S k \) ∼ = ∼ = πk−1 \(GLn \(C\)\) ∼ = ∼ = πk−1 \(Un \), (Daniel Dugger 76) - The stable value in the last row turns out to be 0, although one cannot figure this out without computing a connecting homomorphism in the long exact homotopy sequence (Daniel Dugger 76) - We can write the stable value as πi U where U is the infinite unitary group defined as the colimit of U1 ,→ U2 ,→ U3 ,→ · · · Bott computed the homotopy groups of U to be 2-fold periodic, with πi U = \( Z if i is odd 0 if i is even\. (Daniel Dugger 76) - For a compact and Hausdorff space X, let KO\(X\) denote the Grothendieck group of real vector bundles over X\. Swan’s Theorem gives that KO\(X\) ∼ = ∼ = Kalg \(C\(X\)\), where the latter denotes the Grothendieck group of finitely-generated projectives (Daniel Dugger 77) - We can repeat this definition for both complex and quaternionic bundles, to define groups KU \(X\) and KSp\(X\), respectively\. The group KU \(X\) is most commonly just written K\(X\) for brevity (Daniel Dugger 77) - Remark 13\.4\. Both KO st \(X\) and g KO g \(X, x\) appear often in algebraic topology, and topologists are somewhat cavalier about mixing them up\. (Daniel Dugger 77) - Finally, here is a third description of KO st \(X\)\. Consider the chain of maps Vect0 \(X\) ⊕1 −→ Vect1 \(X\) ⊕1 −→ Vect2 \(X\) ⊕1 −→ · · · The colimit is clearly the set of equivalence classes described in the preceding paragraph, and therefore coincides with KO st \(X\)\. (Daniel Dugger 78) - Let Gr∞ \( R ∞ \) denote the colimit of these maps Gr1 \(R ∞ \) ⊕1 −→ Gr2 \(R ∞ \) ⊕1 −→ Gr3 \(R ∞ \) ⊕1 −→ · · · \(we really want the homotopy colimit, if you know what that is, but in this case the colimit has the same homotopy type and is good enough\)\. (Daniel Dugger 78) - You might recall that Grn \(R ∞ \) is also called BOn , and likewise Gr∞ \(R ∞ \) is also called BO\. (Daniel Dugger 78) - So we have learned that KO st \(X\) ' [X, BO]\. (Daniel Dugger 78) - The calculations of Bott therefore give us the values of g KO g \(S KO k \)\. For k = 0 observe that KO\(S 0 \) = KO\(∗ t ∗\) ∼ = ∼ g \(S 0 \) ∼ = Z ⊕ Z, so we have KO = Z\. This lets us fill out the table: Table 13\.4\. Reduced KO-theory of spheres k 0 1 2 3 4 5 6 7 8 9 10 11 ··· g KO \(S g k \) Z Z /2 Z /2 0 Z 0 0 0 Z Z /2 Z /2 0 ··· (Daniel Dugger 78) - Applying this in particular to X = S k we have that for k ≥ 1 KO g \(S KO k \) ∼ = ∼ = KO st \(S k \) ∼ = ∼ = [S k , BO] ∼ = ∼ = [S k , BO]∗ = πk \(BO\) = πk−1 \(O\)\. (Daniel Dugger 78) - Theorem 13\.5 \(Bott Periodicity, Strong version\)\. There is a weak equivalence of spaces Z × BO ' Ω8 \( Z × BO\)\. (Daniel Dugger 79) - Using Bott Periodicity we can then calculate that for every pointed space X one has g KO g \(Σ KO 8 X\) = [Σ 8 X, Z × BO]∗ = [X, Ω 8 \( Z × BO\)]∗ = [X, Z × BO]∗ = KO g KO g \(X\)\. Remark 13\.6\. In the complex case, Bott Periodicity gives the weak equivalence Z × BU ' Ω2 \( Z × BU \)\. Consequently one obtains K e K\(Σ 2 X\) ∼ = ∼ = K e K\(X\) e for all pointed spaces X\. (Daniel Dugger 79) - Homotopy classes of maps into a fixed space Z always give rise to exact sequences: Proposition 13\.8\. Let X, Y be pointed spaces, and let f : X → Y be a pointed map\. Consider the mapping cone Cf and the natural map p : Y → Cf \. For any pointed space Z, the sequence of pointed sets [X, Z]∗ ← [Y, Z]∗ ← [Cf, Z]∗ is exact in the middle\. (Daniel Dugger 79) - Note that Cj0 ' ΣX and Cj1 ' ΣY \(this is clear from the pictures\)\. Up to sign the map Cj0 → Cj1 is just Σf , so that the sequence of spaces becomes periodic: X → Y → Cf → ΣX → ΣY → Σ\(Cf \) → Σ 2 X → \. \. \. This is called the Puppe sequence\. Note that the composition of two subsequent maps is null-homotopic, and that every three successive terms form a cofiber sequence\. (Daniel Dugger 80) - Given f : X → Y we form the mapping cone Cf , which comes to us with an inclusion j0 : Y ,→ Cf \. Next form the mapping cone on i, which comes with an inclusion j1 : Cf ,→ Cj0 \. Keep doing this forever to get the sequence of spaces X → Y → Cf → Cj0 → Cj1 → · · · depicted below: Note that Cj0 ' ΣX and Cj1 ' ΣY \(this is clear from the pictures\)\. Up (Daniel Dugger 80) - Now let Z be a fixed space and apply [−, Z]∗ to the Puppe sequence\. We obtain the sequence of pointed sets [X, Z]∗ ← [Y, Z]∗ ← [Cf, Z]∗ ← [ΣX, Z]∗ ← [ΣY, Z]∗ ← [Σ\(Cf \), Z]∗ ← \. \. \. By Proposition 13\.8 this sequence is exact at every spot where this makes sense \(everywhere except at [X, Z]∗ \)\. At the left end this is just an exact sequence of pointed sets, but as one moves to the right at some point it becomes an exact sequence of groups \(namely, at [ΣY, Z]∗ \)\. As one moves further to the right, it becomes an exact sequence of abelian groups by the time one gets to [Σ 2 Y, Z]∗ \. (Daniel Dugger 80) - Definition 13\.9\. An infinite loop space is a space Z0 together with spaces Z1 , Z2 , Z3 , \. \. \. and weak homotopy equivalences Zn ' ΩZn+1 for all n ≥ 0\. Note that if Z is an infinite loop space then we really do get a long exact sequence—infinite in both directions—consisting entirely of abelian groups, having the form · · · ← [Cf, Zi+1 ] ← [X, Zi ]∗ ← [Y, Zi ]∗ ← [Cf, Zi ]∗ ← [X, Zi−1 ] ← · · · (Daniel Dugger 81) - where it is convenient to use the indexing convention Z−n = Ω n Z for n > 0\. This situation is very reminiscent of a long exact sequence in cohomology, so let us adopt the following notation: write E i \(X\) = [X+ , Zi ]∗ = \( [X+ , Zi ]∗ i ≥ 0, [Σ −i \(X+ \), Z0 ]∗ i < 0\. For an inclusion of subspaces j : A ,→ X write E i Z \(X, A\) = [Cj, Zi ]+ = \( [Cj, Zi ]∗ i ≥ 0, [Σ i \(Cj\), Z0 ]∗ i < 0\. It is not hard to check that this is a generalized cohomology theory (Daniel Dugger 81) - So we get a generalized cohomology theory whenever we have an infinite loop space\. \(You may know that it works the other way around, too: every generalized cohomology comes from an infinite loop space (Daniel Dugger 81) - For us the importance of all of this is that by Bott’s theorem we have Z × BO ' Ω 8 \( Z × BO\) ' Ω 16 \( Z × BO\) ' \. \. \. \. Thus, Z × BO is an infinite loop space and the above machinery applies\. We obtain a cohomology theory KO ∗ \. Moreover, periodicity gives us that KO i+8 \(X, A\) ∼ = ∼ = KO i \(X, A\), for any i\. (Daniel Dugger 81) - Let us try to compute KO\( R P 2 \)\. (Daniel Dugger 81) - The point of this section was to construct the cohomology theories KO and K, having the properties that when X is compact and Hausdorff the groups KO 0 \(X\) and K 0 \(X\) coincide with the Grothendieck groups of real and complex vector bundles over X\. (Daniel Dugger 81) - Next use the fact that R P 2 can be built by attaching a 2-cell to R P 1 = S 1 , where the attaching map wraps S 1 around itself twice\. That is, R P 2 is the mapping cone for S 1 2 2 −→ S 1 \. (Daniel Dugger 81) - g 0 \(S 1 \) = KOst \(S 1 \) corresponds We have previously seen that the generator of KO to the Mobius bundle [M ] (Daniel Dugger 82) - g \(S 2 \) = KOst \(S 2 \) is [O\(1\)], and the generator of KO the rank 2 bundle whose clutching map is the isomomorphism S 1 → SO\(2\)\. (Daniel Dugger 82) - We happen to know one bundle on RP 2 , the tautological line bundle γ\. (Daniel Dugger 82) - We need g to decide if 2[γ] = 0 in KOst \(RP 2 \); if it is, then KO g \(RP KO 2 \) ∼ = \(Z/2\)2 and if it is g \(RP not then KO \) 2 ∼ = Z/4\. So the question becomes: is γ ⊕ γ stably trivial? (Daniel Dugger 82) - The answer turns out to be that γ ⊕ γ is not stably trivial; this is an elementary exercise using characteristic classes \(Stiefel-Whitney classes\), (Daniel Dugger 82) - Note that this calculation demonstrates an important principle to keep in mind: often the machinery of cohomology theories get you a long way, but not quite to the end, and one has to do some geometry to complete the calculation (Daniel Dugger 82) - Recall that [E] in KOst \(RP 2 \) corresponds to [E] − rank\(E\) in KO g KO g \(RP 2 \); so the class we wrote as [γ] is [γ] − 1 in the shifted perspective, and we need to decide if 2\([γ] − 1\) = 0 in KO\(RP 2 \)\. The element 1 − [γ] should be thought of as corresponding to a chain complex of vector bundles 0 → γ → 1 → 0, and thinking of it this way one finds that it plays the role of the K-theoretic fundamental class of the submanifold RP 1 ,→ RP 2 \. Then \(1 − [γ]\)2 represents the self-intersection product of RP 1 inside RP 2 , which we know is a point by the standard geometric argument \(shown in the picture below, depicting an RP 1 and a small perturbation of it\)\): (Daniel Dugger 82) - For the moment just get the idea that it is the intersection theory of submanifolds in RP g \(RP that is ultimately forcing KO 2 \) to be Z/4 rather than \(Z/2\)2 (Daniel Dugger 83) - Remark 13\.12\. It seems worth pointing out that in fact for every n one has n \) ∼ = Z/2k g \(RP n \) ∼ KO = Z/2k for a certain value k depending on n\. (Daniel Dugger 83) - Exercise 13\.13\. It is a good idea for the reader to try his or her hand at similar calculations, to see how the machinery is working\. Try calculating some of the groups below, at least for small values of n: • K\(CP n \) \(reasonably easy\) • KO\(CP n \) \(a little harder\) • K\(RP n \) \(even harder\) • KO\(RP n \) \(hardest\)\. (Daniel Dugger 83) - It is a classical problem to determine how many independent vector fields one can construct on a given sphere S n \. (Daniel Dugger 83) - This problem was heavily studied throughout the 1940s and 1950s, and then finally solved by Adams in 1962 using K-theory\. (Daniel Dugger 83) - It is one of the great successes of generalized cohomology theories\. (Daniel Dugger 83) - 14\.1\. The vector field problem\. Given a nonzero vector u = \(x, y\) in R 2 , there is a formula for producing a \(nonzero\) vector that is orthogonal to u: namely, \(−y, x\)\. However, there is no analog of this that works in R 3 \. That is, there is no single formula that takes a vector in R 3 and produces a \(nonzero\) orthogonal vector\. If such a formula existed then it would give a nonvanishing vector field on S 2 , and we know that such a thing does not exist by elementary topology\. (Daniel Dugger 83) - Let us next consider what happens in R4 \. Given u = \(x1 , x2 , x3 , x4 \), we can produce an orthogonal vector via the formula \(−x2 , x1 , −x4 , x3 \)\. But of course this (Daniel Dugger 83) - is not the only way to accomplish this: we can vary what pairs of coordinates we choose to flip\. In fact, if we consider −x 2 −x3 −x4 x1 x4 −x3 v 1 = −x4 , v2 = x1 , v3 = x2 \. −x 4 , v2 = x1 , v3 = x2 x 3 −x2 x1 then we find that v1 , v2 , and v3 are not only orthogonal to u but they are orthogonal to each other as well\. In particular, at each point of S 3 we have given an orthogonal basis for the tangent space\. (Daniel Dugger 84) - Question 14\.2\. On S n , how many vectors fields v1 , v2 , \. \. \. , vr can we find so that v 1 , v2 , \. \. \. , vr are linearly independent for each x ∈ S n ? (Daniel Dugger 84) - Note that by the Gram-Schmidt process we can replace “linearly independent” by “orthonormal\.” If n is even, the answer is zero because there does not exist even a single nonvanishing vector field on an even sphere\. (Daniel Dugger 84) - Let u ∈ S 5 have the standard coordinates\. We notice that the vector v1 = \(−x 2 , x1 , −x4 , x3 , −x6 , x5 \) is orthogonal to v\. However, a little legwork shows that no other pattern of switching coordinates will produce a vector that is orthogonal to both u and v1 \. Of course this does not mean that there isn’t some more elaborate formula that would do the job, but it shows the limits of what we can do using our naive constructions\. (Daniel Dugger 84) - For v ∈ S 7 we can divide the coordinates into the top four and the bottom four\. Take the construction that worked for S 3 and repeat it simultaneously in the top and bottom coordinates—this yields a set of three orthonormal vector fields on S 7 , given by the formulas \(14\.2\) \(−x2 , x1 , −x4 , x3 , −x6 , x5 , −x8 , x7 \), \(−x 3 , x4 , x1 , −x2 , −x7 , x8 , x5 , −x6 \), \(−x 4 , −x3 , x2 , x1 , −x8 , −x7 , x6 , x5 \)\. This idea generalizes at once to prove the following: (Daniel Dugger 84) - Proposition 14\.3\. If there exist r \(independent\) vector fields on S n−1 , then there also exist r vector fields on S kn−1 for all k\. (Daniel Dugger 84) - For example, since there is one vector field on S 1 we also know that there is at least one vector field on S 2k−1 for every k\. (Daniel Dugger 84) - Likewise, since there are three vector fields on S 3 we know that there are at least three vector fields on S 4k−1 for every k\. (Daniel Dugger 84) - Recall that S 3 is a Lie group, being the unit quaternions inside of H\. We can choose an orthonormal frame at the origin and then use the group structure to push this around to any point, (Daniel Dugger 84) - hereby obtaining three independent vector fields; in other words, for any point x ∈ S 3 use the derivative of right-multiplication-by-x to transport our vectors in T 1 S 3 to Tx S 3 \. The space S 7 is not quite a Lie group, but it still has a multiplication coming from being the set of unit octonions\. The multiplication is not associative, but this is of no matter—the same argument works to construct 7 vector fields on S 7 \. Note that this immediately gives us 7 vectors fields on S 15 , S 23 , etc\. (Daniel Dugger 85) - Based on the data so far, one would naturally guess that if n = 2r then there are n − 1 vector fields on S n−1 \. However, this guess turns out to fail already when n = 16 \(and thereafter\)\. (Daniel Dugger 85) - To give a sense of how the numbers grow, we give a chart showing the maximum number of vector fields known to exist on low-dimensional spheres: n 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 n−1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 v\.f\. on S n−1 1 3 1 7 1 3 1 8 1 3 1 7 1 3 1 9 (Daniel Dugger 85) - Definition 14\.4\. If n = m · 2a+4b where m is odd, then the Hurwitz-Radon number for n is ρ\(n\) = 2a + 8b − 1\. Theorem 14\.5 \(Hurwitz-Radon\)\. There exist at least ρ\(n\) independent vector fields on S n−1 \. (Daniel Dugger 85) - We will prove the Hurwitz-Radon theorem by a slick, modern method using Clifford algebras\. (Daniel Dugger 85) - 14\.6\. Sums-of-squares formulas\. Hurwitz and Radon were not actually thinking about vector fields on spheres\. They were instead considering an algebraic question about the existence of certain kinds of “composition formulas” for quadratic forms\. For example, the following identity is easily checked: \(x21 + x22 \) · \(y12 + y22 \) = \(x1 y1 − x2 y2 \)2 + \(x1 y2 + x2 y1 \)2 \. Hurwitz and Radon were looking for more formulas such as this one, for larger numbers of variables: (Daniel Dugger 85) - Definition 14\.7\. A sum-of-squares formulas of type [r, s, n] is an identity \(x21 + x22 + \. \. \. + x2r \)\(y12 + y22 + \. \. \. + ys2 \) = z12 + z22 + \. \. \. + zn2 in the polynomial ring R[x1 , \. \. \. , xr , y1 , \. \. \. , ys ], where each zi is a bilinear expression in x’s and y (Daniel Dugger 86) - We will often just refer to an “[r, s, n]-formula”, for brevity\. For what values of r, s, and n does such a formula exist? (Daniel Dugger 86) - This is currently an open question\. There are three formulas that are easily produced, coming from the normed algebras C, H, and O\. The multiplication is a bilinear pairing, and the identity |xy|2 = |x|2 |y|2 is the required sums-of-squares formula\. (Daniel Dugger 86) - Perhaps surprisingly, most of what is known about the non-existence of sumsof-squares formulas comes from topology\. To phrase the question differently, we are looking for a function φ : R r ⊗ R s → R n such that |φ\(x, y\)|2 = |x|2 · |y|2 for all x ∈ R r and y ∈ R s \. The bilinear expressions z1 , \. \. \. , zn are just the coordinates of φ\(x, y\)\. (Daniel Dugger 86) - Corollary 14\.9\. If an [r, n, n]-formula exists, then there exist r − 1 independent vector fields on S n−1 \. (Daniel Dugger 86) - Therefore we have established that φ\(e2 , −\), φ\(e 3 , −\), \. \. \. , φ\(er , −\) are orthonormal vector fields on S n−1 \. (Daniel Dugger 87) - 14\.10\. Clifford algebras\. We have seen that we get r − 1 independent vector fields on S n−1 if we have a sums-of-squares formula of type [r, n, n]\. Having such a formula amounts to producing matrices A2 , A3 , \. \. \. , Ar ∈ On such that A2i = −I and Ai Aj + Aj Ai = 0 for i 6= j\. If we disregard the condition that the matrices be orthogonal, we can encoded the latter two conditions by saying that we have a representation of a certain algebra: Definition 14\.11\. The Clifford algebra Clk is defined to be the quotient of the tensor algebra Rhe1 , \. \. \. , ek i by the relations e2i = −1 and ei ej + ej ei = 0 for all i 6= j\. (Daniel Dugger 87) - The first few Clifford algebras are familiar: Cl0 = R, Cl1 = C, and Cl2 = H\. After this things become less familiar: for example, it turns out that Cl3 = H × H \(we will see why in just a moment\)\. It is somewhat of a miracle that it is possible to write down a precise description of all of the Clifford algebras, and all of their modules\. Before doing this, let us be clear about why we are doing it: Theorem 14\.12\. An [r, n, n]-formula exists if and only if there exists a Clr−1 module structure on R n \. Consequently, if there is a Clr−1 -module structure on R n then there are r − 1 independent vector fields on S n−1 \. (Daniel Dugger 87) - The collection of monomials e i 1 · · · eir for 1 ≤ i1 < i2 < · · · < ir ≤ k give a vector space basis for Clk , which has size 2k (Daniel Dugger 87) - But a miracle now occurs, in that we can analyze all the Clifford algebras by a simple trick\. (Daniel Dugger 88) - To do this part of the argument, we need a slight variant on our Clifford algebras\. Given a real vector space V and a quadratic form q : V → R, define Cl\(V, q\) = TR \(V \)/hv ⊗ v = q\(v\) · 1 | v ∈ V i\. For R with k q\(x1 , \. \. \. , xk \) = −\(x21 + · · · + x2k \) this recovers the algebra Clk \. For q\(x 1 , \. \. \. , xk \) = x21 + · · · + x2k this gives a new algebra we will call Cl − k \. (Daniel Dugger 88) - Proposition 14\.14\. There are isomorphisms of algebras Cl ± ∼ k = ∼ ± = Cl2 ∓ ⊗R Clk−2 \. (Daniel Dugger 88) - In the analysis that follows we will write A\(n\) for the algebra Mn×n \(A\), whenever A is an algebra\. (Daniel Dugger 88) - Table 14\.15\. Clifford algebras (Daniel Dugger 89) - Lemma 14\.16\. There are isomorphisms H ⊗R C ∼ = C\(2\) and H ⊗R H ∼ = R\(4\)\. (Daniel Dugger 90) - 14\.17\. Modules over Clifford algebras\. Now that we know all the Clifford algebras, it is actually an easy process to determine all of their finitely-generated modules\. We need three facts: • If A is a division algebra then all modules over A are free; • By Morita theory, the finitely-generated modules over A\(n\) are in bijective correspondence with the finitely-generated modules over A\. The bijection sends an A-module M to the A\(n\)-module M n \. • If R and S are algebras then modules over R × S can all be written as M × N where M is an R-module and N is an S-module (Daniel Dugger 91) - Table 14\.18\. Dimensions of Clifford modules (Daniel Dugger 91) - Recall that if Clr−1 acts on R n then there are r − 1 independent vector fields on S n−1 \. (Daniel Dugger 91) - Going down the rows of the above table, we make the following deductions: Cl 1 acts on R2 , therefore we have 1 vector field on S Cl 2 acts on R4 , therefore we have 2 vector field on S (Daniel Dugger 91) - the smallest dimension of a nonzero module over Clr is 2σ\(r\) where σ\(r\) = #{s : 0 < s ≤ r and s ≡ 0, 1, 2, or 4 mod \(8\)} 2 σ\(r\) −1 −1 We know that we can construct r independent vector fields on S 2 \. (Daniel Dugger 92) - We will eventually see, following [ABS], that there is a very direct connection between the groups KO∗ and the module theory of the Clifford algebras\. (Daniel Dugger 92) - 14\.20\. Adams’s Theorem\. So far we have done all this work just to construct collections of independent vector fields on spheres\. The Hurwitz-Radon lower bound is classical, and was probably well-known in the 1940’s\. The natural question is, can one do any better? Is there a different construction that would yield more vector fields than we have managed to produce? People were actively working on this problem throughout the 1950’s\. Adams finally proved in 1962 [Ad2] that the Hurwitz-Radon bound was maximal, and he did this by using K-theory: Theorem 14\.21 \(Adams\)\. There do not exist ρ\(n\) + 1 independent vector fields on S n−1 \. (Daniel Dugger 92) - Proposition 14\.22\. If there are r − 1 vector fields on S n−1 then the projection RP un−1 /RP un−r−1 → RP un−1 /RP un−2 ∼ = ∼ = S un−1 has a section in the homotopy 2k−2 2k−2 category, for every u > n \. n (Daniel Dugger 93) - Define V k \(Rn \) = {\(u1 , \. \. \. , uk \) | ui ∈ R n and u1 , \. \. \. , uk are orthonormal}\. This is called the Stiefel manifold of k-frames in R n \. (Daniel Dugger 93) - Consider the map p 1 : Vk \(Rn \) → S n−1 which sends \(u1 , \. \. \. , un \) 7→ u1 \. There exist r vector fields on S n−1 if and only if there is a section of p1 : Vr+1 \(Rn \) → S n−1 \. (Daniel Dugger 93) - We need a fact from basic topology, namely that there is a cell structure on V k \(Rn \) where the cells look like e i 1 × · · · × eis with n − k ≤ i1 < i2 < · · · < is ≤ n − 1 and s is arbitrary\. We will not prove this here: see Hatcher [Ha, Section 3\.D] (Daniel Dugger 93) - Proposition 14\.23\. If n + 2 > 2k then the n-skeleton of our cell structure on V k \(Rn \) is homeomorphic to RP n−1 /RP n−k−1 \. (Daniel Dugger 93) - Exercise 14\.24\. Use singular cohomology to prove that RP n−1 /RP n−3 → S n−1 does not have a section when n is odd\. Deduce that an even sphere does not have a non-vanishing vector field \(which you already knew\)\. (Daniel Dugger 93) - At this point we have seen that there exist cohomology theories K ∗ \(−\) and KO∗ \(−\)\. We have not proven their existence, but we have seen that their existence falls out as a consequence of the Bott periodicity theorems Ω2 \(Z × BU \) ' Z × BU and Ω8 \(Z × BO\) ' Z × BO\. (Daniel Dugger 94) - To some extent we have a “geometric” understanding of K 0 \(−\) and KO0 \(−\) in n terms of Grothendieck groups of vector bundles\. We also know that any K n \(−\) \(or n \(−\)\) group can be shifted to a K 0 KOn \(−\)\) group can be shifted to a K 0 \(−\) group using the suspension isomorphism and Bott periodicity (Daniel Dugger 94) - \. One often hears a slogan like “The geometry behind K-theory lies in vector bundles”\. This slogan, however, doesn’t really say very much; our aim will be to do better\. (Daniel Dugger 94) - One way to encode geometry into a cohomology theory is via Thom classes for vector bundles\. Such classes give rise to fundamental classes for submanifolds and a robust connection wth intersection theory\. (Daniel Dugger 94) - The theory of Thom classes begins with the cohomological approach to orientations\. Recall that (Daniel Dugger 94) - Moreover, an orientation on R n determines a generator for H n \(Rn , Rn − 0\) ∼ = ∼ = Z\. (Daniel Dugger 94) - Now consider a vector bundle p : E → B of rank n\. Let ζ : B → E be the zero section, and write E −0 as shorthand for E −im\(ζ\)\. For any x ∈ B let Fx = p−1 \(x\)\. n \(Fx , Fx − 0\) ∼ = Then H n \(Fx , Fx − 0\) ∼ = Z, and an orientation of the fiber gives a generator\. We wish to consider the problem of giving compatible orientations for all the fibers at once; this can be addressed through the cohomology of the pair \(E, E − 0\)\. (Daniel Dugger 94) - Pick a generator UV ∈ H \(EV , EV − 0\) n ∼ = Z\. (Daniel Dugger 94) - For a neighborhood V of x, let EV = E|V = p−1 \(V \)\. (Daniel Dugger 94) - We think of UV as orienting all of the fibers simultaneously\. (Daniel Dugger 95) - Definition 15\.1\. Given a rank n bundle E → B, a Thom class for E is an element UE ∈ H n \(E, E − 0\) such that for all x ∈ B, jx∗ \(UE \) is a generator in H n \(Fx , Fx − 0\)\. \(Here jx : Fx ,→ E is the inclusion of the fiber\)\. (Daniel Dugger 95) - There is no guarantee that a bundle has a Thom class\. Indeed, consider the following example: (Daniel Dugger 95) - If a bundle E → B has a Thom class then the bundle is called orientable\. Said differently, an orientation on a vector bundle E → B is simply a choice of Thom class in H n \(E, E − 0; Z\)\. (Daniel Dugger 96) - One can also talk about Thom classes with respect to the cohomology theories H ∗ \(−; R\) for any ring R\. Typically one only needs R = Z and R = Z/2, however\. In the latter case, note that any n-dimensional real vector space V has a canonical orientation in H n \(V, V − 0; Z/2\)\. It follows that local Thom classes always patch together to give global Thom classes, and so every vector bundle has a Thom class ∗ in H ∗ \(−; Z/2\)\. (Daniel Dugger 96) - Finally, note that we can repeat all that we have done for complex vector spaces and complex vector bundles\. However, a complex vector space V of dimension n has a canonical orientation on its underlying real vector space, and therefore a canonical generator in H 2n \(V, V − 0\)\. Just as in the last paragraph, this implies that local Thom classes always patch together to give global Thom classes; so every complex vector bundle has a Thom class\. (Daniel Dugger 96) - Theorem 15\.3\. \(a\) Every complex bundle E → B of rank n has a Thom class in H 2n \(E, E − 0\)\. \(b\) Every real bundle E → B of rank n has a Thom class in H n \(E, E − 0; Z/2\)\. (Daniel Dugger 96) - Theorem 15\.4 \(Thom Isomorphism Theorem\)\. Suppose that p : E → B has a Thom class UE ∈ H ∗ \(E, E − 0\)\. Then the map H ∗ \(B\) → H ∗ \(E, E − 0\) given by z 7→ p∗ \(z\) ∪ UE is an isomorphism of graded abelian groups that increases degrees by n\. (Daniel Dugger 96) - 15\.5\. Thom spaces\. The relative groups H ∗ \(E, E − 0\) coincide with the reduced cohomology groups of the mapping cone of the inclusion E − 0 ,→ E\. This mapping cone is sometimes called the Thom space of the bundle E → B, (Daniel Dugger 97) - Definition 15\.6\. Suppose that E → B is a bundle with an inner product\. Define the disk bundle of E as D\(E\) = {v ∈ E | hv, vi ≤ 1}, and the sphere bundle of E as S\(E\) = {v ∈ E | hv, vi = 1}\. (Daniel Dugger 97) - This diagram shows that E − 0 ,→ E and S\(E\) ,→ D\(E\) have weakly equivalent mapping cones (Daniel Dugger 97) - Unlike E −0 ,→ E, however, the map S\(E\) ,→ D\(E\) is a cofibration \(under the mild condition that X is cofibrant, say\): so the mapping cone is weakly equivalent to the quotient D\(E\)/S\(E\)\. This quotient is what is most commonly meant by the term ‘Thom space’: Definition 15\.7\. For a bundle E → B with inner product, the Thom space of E is Th E = D\(E\)/S\(E\)\. (Daniel Dugger 97) - Note that if B is compact then Th E is homeomorphic to the one-point compactification of the space E\. (Daniel Dugger 97) - To see this it is useful to first compactify all the fibers separately, which amounts to forming the pushout of B ← S\(E\) → D\(E\)\. (Daniel Dugger 97) - k \) ∼ = RP n+k /RP n−1 Example 15\.9\. We will show that Th\(nL → RP k \) ∼ = RP n+k /RP n−1 , where L is the tautological line bundle\. (Daniel Dugger 97) - Proposition 15\.13\. For any real bundle E → X one has Th\(E ⊕ n\) ∼ = Σn Th\(E\)\. ∼ 2n = Σ For a complex bundle E → X one has Th\(E ⊕ n\) ∼ 2n = Σ Th\(E\)\. (Daniel Dugger 99) - 15\.12\. Thom spaces for virtual bundles\. Thom spaces behave in a very simple way in relation to adding on trivial bundles: (Daniel Dugger 99) - From this one readily sees that D\(E ⊕ n\)/S\(E ⊕ n\) ∼ = [D\(E\)/S\(E\)] ∧ [Dn /S n−1 ] ∼ = Th\(E\) ∧ S n (Daniel Dugger 99) - Proposition 15\.13 allows one to make sense of Thom spaces for virtual bundles, provided that we use spectra\. (Daniel Dugger 99) - these Thom spectra play a large role in modern algebraic topology\. (Daniel Dugger 99) - 15\.14\. An application to stunted projective spaces\. To demonstrate the usefulness of Thom spaces we give an application to periodicities amongst stunted projective spaces\. This material will be needed later, in the solution of the vector fields on spheres problem\. (Daniel Dugger 100) - The natural question arises: fixing a and b, what values of r \(if any\) satisfy Σ r [RP a+b /RP a ] ' RP a+b+r /RP a+r \. (Daniel Dugger 100) - One can use singular cohomology and Steenrod operations to produce some necessary conditions here\. For example, integral singular homology easily yields that if if b ≥ 2 then r must be even\. Use of Steenrod operations produces more stringent conditions \( (Daniel Dugger 100) - We will use Thom spaces to provide some sufficient conditions for a stable homotopy equivalence between stunted projective spaces (Daniel Dugger 100)