# 02/09/2019: Characteristic Classes (John Milnor & James D. Stasheff) /home/zack/Dropbox/Library/John Milnor/Characteristic Classes (654)/Characteristic Classes - John Milnor.pdf Last Annotation: 02/09/2019 ## Notes - Definition of smooth or C^\infty (John Milnor & James D. Stasheff 7) - Definition of a smooth manifold (John Milnor & James D. Stasheff 8) - Expression of tangent space as the span of partial derivatives (John Milnor & James D. Stasheff 10) - The derivative is an endofunctor on the category of smooth manifolds (John Milnor & James D. Stasheff 12) - Definition of a vector bundle (John Milnor & James D. Stasheff 17) - Local triviality condition (John Milnor & James D. Stasheff 17) - Definition: Local coordinate system (John Milnor & James D. Stasheff 17) - Definition: Isomorphism of vector bundles (John Milnor & James D. Stasheff 18) - Definition: the normal bundle (John Milnor & James D. Stasheff 19) - Definition of the canonical line bundle in RP^n (John Milnor & James D. Stasheff 19) - Theorem: The canonical line bundle is nontrivial for n >= 1 (John Milnor & James D. Stasheff 20) - Definition: Section of a vector bundle (John Milnor & James D. Stasheff 20) - Definition: a nowhere zero section (John Milnor & James D. Stasheff 20) - The canonical line bundle on RP^n does not have a nowhere zero section (John Milnor & James D. Stasheff 20) - Definition: Independent sections (John Milnor & James D. Stasheff 22) - An isomorphism of total spaces that isomorphically maps fibers to fibers is an isomorphism of bundles (John Milnor & James D. Stasheff 22) - Proof that S^3 is parallelizable\. (John Milnor & James D. Stasheff 24) - Definition: Quadratic function (John Milnor & James D. Stasheff 25) - Deriving an inner product from a quadratic map (John Milnor & James D. Stasheff 25) - Trivial bundle iff there exist n independent orthonormal sections (John Milnor & James D. Stasheff 26) - Definition of a map of bundles (John Milnor & James D. Stasheff 30) - Definition: Whitney Sums (John Milnor & James D. Stasheff 31) - Definition: subbundle (John Milnor & James D. Stasheff 31) - Bundles split as the Whitney sum of any subbundle and its perp (John Milnor & James D. Stasheff 32) - Definition: Immersion (John Milnor & James D. Stasheff 34) - Big list of ways to combine vector spaces (John Milnor & James D. Stasheff 35) - Definition of a continuous functor (John Milnor & James D. Stasheff 36) - Definition: The tensor product of bundles (John Milnor & James D. Stasheff 37) - Definition: Submersion (John Milnor & James D. Stasheff 39) - Axioms for Stiefel-Whitney classes (John Milnor & James D. Stasheff 41) - Definition: The Stiefel-Whitney numbers (John Milnor & James D. Stasheff 54) - How to compute a Stiefel-Whitney number (John Milnor & James D. Stasheff 55) - Stiefel-Whitney numbers classify manifolds up to cobordism (John Milnor & James D. Stasheff 57) - Theorem: The cohomology ring of the infinite Grassmanian in Z/2Z coefficients is generated by the Stiefel-Whitney classes (John Milnor & James D. Stasheff 85) - Definition: The Thom Isomorphism $$Needed to define the Stiefel-Whitney class$$ (John Milnor & James D. Stasheff 92) - Definition of the fundamental class (John Milnor & James D. Stasheff 92) - Definition: Stiefel-Whitney Class $$Depends on Thom's identity$$ (John Milnor & James D. Stasheff 93) - Definition: Orientation of a bundle (John Milnor & James D. Stasheff 98) - Definition: The Euler class of an n-plane bundle (John Milnor & James D. Stasheff 99)
## Highlights - DEFINITION\. A real vector bundle £ over B consists of the following: 1\) a topological space E = E$$£$$ called the total space, 2\) a $$continuous$$ map 7: E » B called the projection map, and 3\) for each b ¢ B the structure of a vector space” over the real numbers in the set 7~1$$b$$\. (John Milnor & James D. Stasheff 17) - These must satisfy the following restriction: Condition of local triviality\. For each point b of B there should exist a neighborhood U C B, an integer n> 0, and a homeomorphism h:Ux RP > 71\) so that, for each b ¢ U, the correspondence x + h$$b, x$$ defines an isomorphism between the vector space R™ and the vector space 7 1$$b$$\. (John Milnor & James D. Stasheff 17) - Such a pair $$U,h$$ will be called a local coordinate system for & about b\. If it is possible to choose U equal to the entire base space, then & will be called a trivial bundle\. (John Milnor & James D. Stasheff 17) - In Steenrod’s terminology an R™-bundle is a fiber bundle with fiber R™ and with the full linear group GL$$R$$ in n variables as structural group\. (John Milnor & James D. Stasheff 18) - Example 2\. The tangent bundle ry; of a smooth manifold M\. (John Milnor & James D. Stasheff 18) - If ry is a trivial bundle, then the manifold M is called parallelizable\. (John Milnor & James D. Stasheff 19) - THEOREM 2\.1\. The bundle Ve over P" is not trivial, for n> 1\. (John Milnor & James D. Stasheff 20) - $$A cross-section of the tangent bundle of a smooth manifold M is usually called a vector field on M\.$$ (John Milnor & James D. Stasheff 20) - THEOREM 2\.2\. An R™-bundle ¢ is trivial if and only if & admits n cross-sections s,,\.\.\.,S, n which are nowhere dependent\. n (John Milnor & James D. Stasheff 22) - Hence S° is parallelizable\. (John Milnor & James D. Stasheff 24) - DEFINITION\. A Euclidean vector bundle is a real vector bundle ¢& together with a continuous function §: E > R such that the restriction of pu to each fiber of ¢ is positive definite and quadratic\. The function pu itself will be called a Euclidean metric on the vector bundle ¢£\. (John Milnor & James D. Stasheff 25) - In the case of the tangent bundle ry; of a smooth manifold, a Euclidean metric yp: DM -> R (John Milnor & James D. Stasheff 25) - is called a Riemannian metric, and M together with pu is called a Riemannian manifold\. (John Milnor & James D. Stasheff 26) - ote\. In Steenrod’s terminology a Euclidean metric on & gives-rise to a reduction of the structural group of £ from the full linear group to the orthogonal group\. (John Milnor & James D. Stasheff 26) - A priori there appear to be two different concepts of triviality for Euclidean vector bundles; (John Milnor & James D. Stasheff 26) - Show that the unit sphere S" admits a vector field which is nowhere zero, providing that n is odd\. Show that the normal bundle of S\$ c R™! is trivial for all n\. (John Milnor & James D. Stasheff 27) - If S" admits a vector field which is nowhere zero, show that the identity map of S" is homotopic to the antipodal map\. (John Milnor & James D. Stasheff 27) - For n even show that the antipodal map of S™ is homotopic to the reflection {CSP Xn41\) = $$=X, Xy, 000, Xp$$ (John Milnor & James D. Stasheff 27) - and therefore has degree —1\. (John Milnor & James D. Stasheff 27) - show that S? is not parallelizable for n even, n> 2\. (John Milnor & James D. Stasheff 27) - More generally a smooth map f:M -» N between smooth manifolds is called an immersion if the Jacobian Df, : DM, -> DN¢$$x$$ maps the tangent space DM, injectively $$i\.e\., with kernel zero$$ for each x ¢ M\. [It follows from the implicit function theorem that an immersion is locally an embedding of M in N, but in the large there may be selfintersections\. (John Milnor & James D. Stasheff 34) - if M C N with normal bundle v, where N is a smooth Riemannian manifold, then the ‘second fundamental form’’ can be defined as a smooth symmetric cross-section of the bundle Hom $$ry; ® ney$$ (John Milnor & James D. Stasheff 39) - A module is projective if it is a direct summand of a free module\. (John Milnor & James D. Stasheff 40) - THEOREM 4\.9 [Pontrjagin]\. If B is a smooth compact $$n+1$$dimensional manifold with boundary equal to M $$compare §17$$, then the Stiefel-Whitney numbers of M are all zero\. (John Milnor & James D. Stasheff 56) - THEOREM 4\.10 [Thom]\. If all of the Stiefel-Whitney numbers of M are zero, then M can be realized as the boundary of some smooth compact manifold\. (John Milnor & James D. Stasheff 57) - DEFINITION\. Two smooth closed n-manifolds M; and M, belong to the same unoriented cobordism class iff their disjoint union M; UM, is the boundary of a smooth compact $$n+l$$-dimensional manifold\. (John Milnor & James D. Stasheff 57) - DEFINITION\. The Grassmann manifold G$$Ro+K$$ is the set of all n-dimensional planes through the origin of the coordinate space ROK, (John Milnor & James D. Stasheff 60) - An n-frame in R™K is an n-tuple of linearly independent vectors of R™K, (John Milnor & James D. Stasheff 60) - The collection of all n-frames in R™¥ forms an open subset of the n-fold Cartesian product ROHK X eee X RO+K called the Stiefel manifold v$$ROK$$, (John Milnor & James D. Stasheff 60) - THEOREM 7\.1\. The cohomology ring H\*G,; 2/2\) is a polynomial algebra over 7/2 freely generated by the Stiefel-Whitney classes w, $$YD, ery w,\(y™$$\. (John Milnor & James D. Stasheff 85) - PROPERTY 9\.7\. If the oriented vector bundle ¢ possesses a nowhere zero cross-section, then the Euler class e$$£$$ must be zero\. (John Milnor & James D. Stasheff 103) - LEMMA 14\.1\. If w is a complex vector bundle, then the underlying real vector bundle wg has a canonical preferred orientation\. (John Milnor & James D. Stasheff 154) - Applying this lemma to the special case of a tangent bundle, it follows that any complex manifold has a canonical preferred orientation\. (John Milnor & James D. Stasheff 154) - every orientation for the tangent bundle of a manifold gives rise to a unique orientation of the manifold\. (John Milnor & James D. Stasheff 154) - But we already know from Section 14 that any characteristic class for complex vector bundles can be expressed as a polynomial in the Chern classes\. (John Milnor & James D. Stasheff 294)