## Highlights - For many years, Algebraic Topology rests on three legs: “ordinary” Cohomology, K-theory, and Cobordism (Budyak 6) - The pioneering work of Pontryagin and Thom forged a deep connection between certain geometric problems \(such as the classiﬁcation of manifolds\) and homotopy theory, through the medium of the Thom space (Budyak 6) - Computations become possible upon stabilization, and this provided some of the ﬁrst and most compelling examples of “spectra\.” (Budyak 6) - This international tradition was continued with the more or less simultaneous work by Novikov and Milnor on complex cobordism, and later by Quillen\. More recently Dennis Sullivan opened the way to the study of “manifolds with singularities,” a study taken up most forcefully by the Russian school, notably by Vershinin, Botvinnik, and Rudyak\. (Budyak 6) - There is a ﬁne introduction to the stable homotopy category\. The subtle and increasingly important issue of phantom maps is addressed here with care\. Equally careful is the treatment of orientability (Budyak 6) - The contents of this book are concentrated around Thom spaces \(spectra\), orientability theory and \(co\)bordism theory \(including \(co\)bordism with singularities\), framed by \(co\)homology theories and spectra\. (Budyak 10) - we consider the \(inter\)connections between geometry and homotopy theory, since Thom spectra and related matters are now the main tools for this interplay (Budyak 11) - of the ﬁrst results in this area was the Gauss–Bonnet formula, relating a geometrical invariant \(the curvature\) to a homotopical one \(the Euler characteristic\)\. Proceeding, we can recall the Riemann–Roch Theorem, the Poincaré integrality theory, relationships between critical points of a smooth function on a smooth manifold and its homotopy type \(Lusternik– Schnirelmann, Morse\), the de Rham Theorem, etc\. (Budyak 11) - The term “cohomology theory” is used for what was previously called “generalized” or “extraordinary” cohomology theory, i\.e\., for functors which satisfy all the Eilenberg– Steenrod axioms except the dimension axioms\. (Budyak 11) - Every homology theory h∗ \(−\) yields a so-called dual cohomology theory h ∗ \(−\), and vice versa\. They are connected via the equality h h i \(X\) = h hn−i \(Y \) where Y is n-dual to X \(and tilde denotes the reduced \(co\)homology\)\. (Budyak 11) - Thom spaces\. The Thom space T ξ of a locally trivial Rn -bundle ξ = {p : E → B} is deﬁned as follows\. Let ξ • be the S n -bundle obtained from ξ by the ﬁberwise one-point compactiﬁcation, and let E • be the total space of ξ \. Then the “inﬁnities” of the ﬁbers form a section s : B → E • , and we deﬁne T ξ := E • /s\(B\)\. Furthermore, the Thom space of a spherical ﬁbration {p : E → B} is the cone C\(p\) of the projection (Budyak 12) - p\. For example, the Thom space of the Rn -bundle over a point is S n , the Thom space of the open Möbius band \(considered as the R1 -bundle over S 1 \) is the real projective plane RP 2 , the Thom space of the Hopf bundle S 3 → S 2 \(with ﬁber S 1 \) is the complex projective plane CP 2 \. We use Thom’s notation M On for the Thom space T γ n of the universal n-dimensional vector bundle γ n over the classifying space BOn , i\.e\., M On := T γ n ; e\.g\., M O1 = RP ∞ (Budyak 12) - \.C\. Whitehead observed the importance of the structure on the normal bundle in classifying structures on manifolds\. It turns out that Thom spaces establish an adequate context for this\. Namely, for every closed smooth manifold M n , the set of \(diﬀeomorphism classes of\) smooth manifolds homotopy equivalent to M is controlled by the group πn+N \(T ν\), where ν is the normal bundle of an embedding of M in Rn+N with N large enough (Budyak 12) - This is closely related to the Milnor–Spanier–Atiyah Duality Theorem, which asserts that T ν and M/∂M are stable N -dual for every compact manifold M \. This theorem clariﬁes connections between manifolds and their normal bundles and enables us to transmit properties of bundles to properties of manifolds\. For example, we have the Thom isomorphism ϕ : H i \(X; Z/2\) → H H i+n \(T ξ; Z/2\) for every locally trivial Rn -bundle ξ over a space X, and the above theorem transforms it to the Poincaré duality H i \(M ; Z/2\) ∼ = ∼ = Hn−i \(M, ∂M ; Z/2\) for every compact n-dimensional manifold M \. (Budyak 12) - Turning to another example, I recall the Thom formula wi \(ξ\) = ϕ −1 Sq i uξ where ξ is an n-dimensional vector bundle over a space X, wi \(ξ\) is its i-th Stiefel–Whitney class, ϕ : H i \(X; Z/2\) → H H i+n \(T ξ; Z/2\) is the Thom isomorphism and uξ ∈ H \(T ξ; Z/2\) is the Thom class of ξ\. This formula expands n a geometric invariant \(the Stiefel–Whitney class\) via the Steenrod operation which is a purely homotopic thing\. Moreover, we can use the formula in order to deﬁne the Stiefel–Whitney classes of spherical ﬁbrations\. In particular, it becomes clear that the Stiefel–Whitney classes are invariants of the ﬁber homotopy type of a vector bundle (Budyak 12) - Generalizing, we can consider an arbitrary natural transformation τ : h ∗ → k ∗ of cohomology theories instead of Sq i \. Then, under suitable conditions on ξ, there is a generalized Thom class u hξ ∈ h h n \(T ξ\) and a generalized Thom isomorphism ϕk : k i \(X\) → k k i+n \(T ξ\), and so we can form the class K\(ξ\) = ϕ −1 k τ u h ξ which is an analogue and generalization of the Stiefel–Whitney class (Budyak 13) - So, we have a large source of invariants of Rn -bundles\. For example, the Todd genus and the A genus are particular cases of this construction\. Moreover, the wellA known integrality theorems which are related to Todd and A genera can be A generalized for the class K (Budyak 13) - Pontrjagin [1] proved that if a manifold bounds then all its characteristic numbers are trivial\. In particular, RP 2 does not bound because w2 \(RP 2 \) = 0\. So, N2 = 0, i\.e\., some groups Nk are non-trivial (Budyak 13) - Well, but how to compute Nk ? Clearly, N0 = Z/2, N1 = 0\. Using the classiﬁcation of closed surfaces, one can prove that N2 = Z/2: every orientable surface bounds, and every non-orientable surface either bounds or is bordant to RP 2 ; and RP 2 does not bound (Budyak 13) - Rokhlin [1] proved that N3 = 0, using complicated and tricky geometry\. The further computation of Nk looked absolutely hopeless; however this was done by Thom [2] via an exciting and successful application of homotopy theory\. Namely, Thom proved that N k = πk+N \(M ON \) for N large enough (Budyak 13) - he answer is N∗ = Z/2 [xi ], dim xi = i, i ∈ N, i = 2s − 1 (Budyak 13) - where N∗ = ⊕Nk is the graded ring with the multiplication induced by the direct product of manifolds (Budyak 13) - The above constructions can be generalized: we can consider oriented manifolds or, more generally, manifolds equipped with some extra structures\. As above, there arise certain bordism groups, and they can be interpreted as homotopy groups of certain Thom spaces (Budyak 14) - We use a category of spectra proposed by Adams [5]\. So, a spectrum E is a sequence {En , sn }∞ n=−∞ of pointed CW -spaces En and pointed CW embeddings sn : SEn → En+1 where S denotes the pointed suspension\. (Budyak 14) - \(1\) For every pointed space X we have the spectrum Σ ∞ X = {S n X, sn } where sn : SS n X → S n+1 X is the identity map\. \(2\) For every pointed space X and every spectrum E = {En , sn } we have the spectrum X ∧ E = {X ∧ En , 1 ∧ sn }\. (Budyak 14) - \(3\) Let θ 1 be the trivial 1-dimensional vector bundle over BOn , and let the map BOn → BOn+1 \(assuming it to be an embedding\) classify the vector bundle γ n ⊕ θ1 \. Then we have a map sn : T \(γ n ⊕ θ1 \) → T γ n+1 \. Moreover, (Budyak 14) - Clearly, Nk = Nk \(pt\)\. Moreover, N k \(X\) = πk+N \(X + ∧ M ON \) for N large enough, where X + is the disjoint union of the space X and a point\. (Budyak 14) - one can prove that T \(γ n ⊕ θ1 \) = ST γ n = SM On , and so we have the Thom spectrum M O = {M On , sn (Budyak 15) - Given a spectrum E, we have the homomorphisms hk,n : πk \(En \) → πk+1 \(SEn \) \(sn \)∗ \(sn \)∗ −−−→ πk+1 \(En+1 \)\. We deﬁne the homotopy group πk \(E\) to be the direct limit of the sequence ··· − → πk+n \(En \) hk+n,n hk+n,n −−−−→ πk+n+1 \(En+1 \) − → ··· , i\.e\., πk \(E\) = lim n→∞ πi+n \(En \)\. (Budyak 15) - N k = πk \(M O\), Nk \(X\) = πk \(X + ∧ M O\) and so get rid of “N large enough” (Budyak 15) - Every spectrum E yields a homology theory E∗ \(−\) and a cohomology theory E ∗ \(−\) by the formulae Ei \(X\) := lim n→∞ πi+n \(X + ∧ En \), E i \(X\) := lim n→∞ [S n X + , Ei+n ]\. Moreover, E∗ \(−\) and E ∗ \(−\) are dual to each other\. Conversely, every \(co\)homology theory can be represented by a spectrum via the above formulae\. (Budyak 15) - Note that, in particular, the spectrum M O yields the bordism \(resp\. cobordism\) theory N∗ \(−\) \(resp\. N∗ \(−\)\)\. (Budyak 15) - Orientability\. We consider in this book orientability with respect to arbitrary cohomology theories (Budyak 15) - Similarly, we deﬁne a locally trivial Rn -bundle ξ over a connected base to be orientable if H n \(T ξ\) = Z, and an orientation of ξ is a generator of the group H n \(T ξ (Budyak 16) - For example, an h-orientation of an Rn -bundle ξ is a suitable element uξ ∈ h h n \(T ξ\), an horientation of a closed manifold M n is an element [M ] ∈ hn \(M n \)\. (Budyak 16) - Furthermore, one can develop an elegant theory of characteristic classes taking values in h∗ \(−\) provided that all complex vector bundles are horientable; these classes generalize the classical Chern classes\. (Budyak 16) - As the last example, we mention that general orientabilty theory provides a formal group input to algebraic topology; this matter is completely degenerate for classical cohomology, and so this remarkable theory was able to appear only under the general approac (Budyak 16) - oach\. (Budyak 16) - \(Co\)bordism with singularities\. \(Co\)bordism with singularities is now a common and convenient notion, being a favorite tool as well as subject of research in algebraic topology\. Roughly speaking, we take a class of manifolds and extend it to a class of suitable polyhedra \(manifolds with singularities\) where a notion of a boundary is reasonably deﬁned (Budyak 16) - Moreover, the famous Morava k-theories are also constructed as certain cobordism with singularities (Budyak 17) - ories are also constructed as certain cobordism with singularities\. I also want to mention an application of \(co\)bordism with singularities to the topological quantum ﬁeld theory: for example, the elliptic \(co\)homology can be constructed as \(co\)bordism with singularities (Budyak 17) - Finally, \(co\)bordism with singularities gives a natural geometric ﬂavor to algebraic or homotopical matters\. For example, the Adams resolution of certain spectra can be interpreted in terms of \(co\)bordism with singularities, and this enables us to get useful information about some classical \(co\)bordism theories, like M SU and M Sp (Budyak 17) - paradigm of algebraic topology, and it freshly demonstrated the power and usefulness of the relations between homotopy theory and geometry\. In order to exhibit relatively rece (Budyak 17) - The paper Thom [2] made a revolution and formed the contemporary paradigm of algebraic topology, and it freshly demonstrated the power and usefulness of the relations between homotopy theory and geometry\. In order to exhibit relatively recent advantages of this matter, I just write down a list \(unavoidably incomplete\) of certain geometric problems which were \(partially or completely\) solved via an application of homotopy theory\. 1 \(1\) When can a manifold M be immersed in a manifold N , and how can one classify these immersions? \(Smale [1], Hirsch [1]\.\) \(2\) When can a homology class in a space be realized by a map of a closed manifold? \(Thom [2]\.\) \(3\) When is a closed manifold a boundary of a compact manifold with boundary? \(Thom [2]\.\) \(4\) Which spaces are homotopy equivalent to closed smooth manifolds? \(Browder [1,2], Novikov [2,3]\.\) \(5\) How can one classify manifolds up to diﬀeomorphism \(PL isomorphism, homeomorphism\)? \(Smale [1], Kervaire–Milnor [1], Browder [1,2], Novikov [2,3], Hirsch–Mazur [1], Sullivan [1], Kirby–Siebenmann [1], Freedman [1], Donaldson [1]\.\) \(6\) How many pointwise linearly independent tangent vector ﬁelds exist on the n-dimensional sphere? \(Adams [3]\.\) \(7\) Which smooth manifolds admit a Riemannian metric of positive scalar curvature? \(Gromov–Lawson [1], Stolz [1]\.\) (Budyak 17) - The category consisting of sets \(as objects\) and functions \(as morphisms\) is denoted by E ns\. The category of pointed sets and pointed functions is denoted by E ns• \. (Budyak 19) - partially ordered set, or a poset, is a quasi-ordered set with the following condition: if λ ≤ μ ≤ λ then λ = μ (Budyak 20) - A chain in a poset is a family {ai } such that, for every pair i, j of indices, either ai ≤ aj or aj ≤ ai \. An upper bound of the chain is any a such that a i ≤ a for every i\. A poset is called inductive if every chain in it has an upper bound\. (Budyak 20) - 1\.3\. Zorn’s Lemma\. Every inductive set has a maximal element\. (Budyak 20) - K → E ns is represented by a certain object B of K if there exists a natural equivalence F ∼ = TB \. In this case B is called a classifying or representing object for F \. Furthermore, F is called representable if it can be represented by some B\. (Budyak 20) - 1\.4\. Deﬁnition\. We say that a contravariant functor F : K → E ns is represented by a certain object B of K if there exists a natu (Budyak 20) - Let F, G : K → E ns be represented by B, C respectively\. It is obvious that every morphism f : B → C yields a natural transformation Tf : TB → TC and hence F → G\. The converse is also true\. ∼ = ∼ = ∼ = ∼ = 1\.5\. Lemma \(Yoneda\)\. Fix natural equivalences b : F −→ TB , c : G −→ TC \. For every natural transformation ϕ : F → G there exists a morphism f : B → C such that for every object X of K the diagram ϕ F \(X\) −−−−→ G\(X\) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ b T f T Tf B \(X\) −−−−→ TC \(X\) commutes, and such a morphism f is unique\. In particular, the representing object B for F is determined by F uniquely up to isomorphism\. (Budyak 20) - The category of abelian groups and homomorphisms is denoted by A G \. Note that the usual direct product of abelian groups is the categorical product in A G , while the usual direct sum is the categorical coproduct in A G \. (Budyak 21) - In algebraic context, we reserve the word “unit” for the neutral element of a monoid \(group\)\. In particular, the multiplicative identity element of a ring is also called the unit\. (Budyak 21) - Given a commutative ring R, we denote by R[x, y, \. \. \. , z] the polynomial ring of indeterminates x, y, \. \. \. , z\. The corresponding power series ring is denoted by R[[x, y, \. \. \. , z]]\. If R is a graded ring, we assume that x, y, \. \. \. , z are homogeneous indeterminates\. Furthermore, ΛR \(x, y, \. \. \. , z\) denotes the free exterior algebra \(with a unit\) over R of indeterminates x, y, \. \. \. , z, and for a graded R we assume that x, y, \. \. \. , z have odd degrees\. We use the notation Λ\(x, y, \. \. \. , z\) for the ring ΛZ \(x, y, \. \. \. , z\)\. (Budyak 21) - Let ρ : A → B be a ring homomorphism, and let M be a right Amodule\. The homomorphism ρ turns B into a left A-module ρ B, where a·b = ρ\(a\)b for a ∈ A, b ∈ B, cf\. Cartan–Eilenberg [1]\. We can therefore form the tensor product over A of A-modules M, B\. This tensor product is denoted by M ⊗ρ B (Budyak 21) - \(a\) Let K be a category\. A direct system over Λ, or brieﬂy, a direct Λsystem, in K is a covariant functor M : Λ → K \. In other words, M is a family M = {Mλ , j λμ }λ , μ ∈ Λ, where Mλ ∈ K and where jλμ : Mμ → Mλ for μ ≤ λ are morphisms such that jλμ jμν = jλν for ν ≤ μ ≤ λ and jλλ = 1Mλ \. \(b\) A morphism f : {Mλ , jλμ } → {Nλ , hμλ } of direct Λ-systems is a natural transformation of functors, i\.e\., a family {fλ : Mλ → Nλ } with hμλ fμ = fλ jλμ \. (Budyak 22) - 2\.4\. Deﬁnition\. Let Λ be a quasi-ordered set, and let {Aλ }λ∈Λ be a direct Λsystem of abelian groups\. Let iλ : Aλ → ⊕λ Aλ be the inclusion, and let B ⊂ ⊕λ Aλ be the subgroup generated by all elements of the form \(iμ aμ − iλ jλμ aμ \)\. The quotient group \(⊕λ Aλ \)/B is called the direct limit of the direct system {Aλ } and is denoted by −→{Aλ }\. lim (Budyak 22) - 2\.5\. Theorem\. Let G be an abelian group, and let ϕλ : Aλ → G be a family of homomorphisms such that ϕμ = ϕλ jλμ for every μ ≤ λ\. Then there exists a homomorphism ϕ : −→{Aλ } → G such that ϕkλ = ϕλ for every λ\. lim (Budyak 22) - Given two pairs \(X, A\), \(Y, B\) of spaces and a map f : A → B, the space X ∪f Y is deﬁned to be the quotient space \(X Y \)/ ∼, where ∼ is the smallest equivalence relation generated by the following relation: a ∼ b if f \(a\) = b for a ∈ A, b ∈ B\. We say that the space X ∪f Y is obtained from X by adjoining, or gluing, Y via f \. For instance, if Y = pt = B then X ∪f Y ∼ = X/A\. (Budyak 24) - A triad \(X; A, B\) is a topological space and two of its closed subspaces A, B such that X = A ∪ B\. A ﬁltration of a topological space X is a sequence {· · · ⊂ X0 ⊂ · · · ⊂ Xn ⊂ · · · ⊂ X} such that: \(1\) X = ∪n Xn \. \(2\) Every Xn is closed in X\. \(3\) X inherits the direct limit topology, i\.e\., U is open in X iﬀ U ∩ Xn is open in Xn for every n\. (Budyak 24) - Given a space X, we denote by X + the disjoint union of X and a point, and the added point is assumed to be the base point\. (Budyak 24) - Algebraic topologists prefer to deal with “nice” spaces, such as CW spaces\. However, a class of spaces in which algebraic topologists work should be closed under standard operations which topologists use\. In other words, the suitable category of spaces should be large enough to accommodate operations and small enough to rule out pathologies at the same time\. One such category was suggested by Steenrod [2] and improved by McCord [1]3 , and is known as the category of weak Hausdorﬀ compactly generated spaces (Budyak 24) - 3\.2\. Deﬁnition\. \(a\) A topological space X is called weak Hausdorﬀ if, for every map ϕ : C → X of a compact space C, the set ϕ\(C\) is closed in X\. (Budyak 24) - \(b\) A subset U of a topological space X is called compactly open if ϕ−1 \(U \) is open for every map ϕ : C → X of a compact space C\. A topological space X is called compactly generated if each of its compactly open sets is open\. (Budyak 25) - Note that every point of a weak Hausdorﬀ space is closed, and that every Hausdorﬀ space is weak Hausdorﬀ\. Thus, the weak Hausdorﬀ property lies between T1 and T2 \. (Budyak 25) - Generally speaking, the usual Cartesian product of two spaces from W is not in W \. See Dowker [1], §5\. Nevertheless, the category W admits products (Budyak 26) - 3\.6\. Deﬁnition\. Given a family {Xi } of topological spaces, we deﬁne their compactly generated direct product c Xi := k Xi i i where is the usual Cartesian product of topological spaces (Budyak 26) - Deﬁne the compact-open topology as follows: let ϕ : C → X be a map of a compact space C, and let U be an open set in Y \. We denote by W \(ϕ, U \) the set of all maps f : X → Y such that f ϕ\(C\) ⊂ U \. Then the family {W \(ϕ, U \)} for all such pairs \(ϕ, U \) forms a subbasis of the compact-open topology on the set of maps from X to Y \. (Budyak 27) - \(c\) The loop space Ω\(X, ∗\) of a pointed space \(X, ∗\) is just the pointed space \(X, ∗\)\(S 1 ,∗\) where S 1 is the circle\. (Budyak 27) - 3\.11\. Convention\. Throughout the book we will assume that all spaces belong to W unless somthing else is said explicitly, i\.e\., the word “space” means “weak Hausdorﬀ compactly generated space”\. (Budyak 27) - 3\.12\. Basic homotopy theory\. \(a\) Two maps f, g : X → Y are called homotopic if there is a map \(homotopy, or deformation\) H : X × I → Y such that H|X × {0} = f and H|X × {1} = (Budyak 28) - \(b\) A map f : X → Y is called a homotopy equivalence if there is a map g : Y → X such that gf 1X and f g 1Y \. In this case we say that f and g are homotopy inverse to each other\. Two spaces X, Y are called homotopy equivalent if there is a homotopy equivalence X → Y , and we write X Y \. The homotopy type of a space X is the class of all spaces homotopy equivalent to X\. (Budyak 28) - \(c\) By saying that two maps f, g : \(X, A\) → \(Y, B\) are homotopic we mean that there exists a homotopy \(X × I, A × I\) → \(Y, B\)\. Furthermore, we say that two maps f, g : \(X, A\) → \(Y, B\) are homotopic relative to A, and write f g rel A, if there is a homotopy H : f g such that H\(a, t\) = f \(a\) for every a ∈ A, t ∈ I (Budyak 28) - 3\.13\. Deﬁnition\. We say that a map is essential if it is not homotopic to a constant map\. (Budyak 28) - 3\.14\. Deﬁnition\. Let H W denote the category whose objects are the same as those of W but whose morphisms are the homotopy classes of maps\. Clearly, every diagram in W yields a diagram in H W \. We say that a diagram in W is homotopy commutative if the corresponding diagram in H W is commutative (Budyak 28) - \(a\) The mapping cylinder, or just the cylinder, of f is the space M f := X × [0, 1] ∪f Y, f where f is considered as the map X × {0} = X − → Y \. Recall that there is a standard deformation F : M f × I → Y where F \(\(x, t\), s\) = \(x, st\) if \(x, t\) ∈ X × \(0, 1] and s > 0 F \(\(x, t\), 0\) = f \(x\) if \(x, t\) ∈ X × \(0, 1] F \(y, s\) = y if y ∈ Y\. Note that F M f × {0} : M f → Y is a retraction and F M f × {1} = 1Mf , i\.e\., Y is a deformation retract of M f \. (Budyak 29) - \(b\) The mapping cone, or the coﬁber, or just the cone, of f is the space Cf := M f /\(X × {1}\)\. (Budyak 29) - 3\.18\. Deﬁnition\. \(a\) Given two maps f : X → Y and g : X → Z, the double f g mapping cylinder of the diagram Y ← −X − → Z is the space D := X × [0, 2] ∪ϕ \(Y Z\) (Budyak 29) - For instance, Cf is \(homeomorphic to\) the double mapping cylinder of f the diagram Y ← −X − → pt\. (Budyak 30) - \(b\) The mapping cone of the constant map X → pt is called the suspension over a space X and denoted by SX (Budyak 30) - \. Thus, the suspension is the double mapping cylinder of the diagram pt ← − X − → p (Budyak 30) - \(c\) The mapping cylinder of the trivial map X → pt is denoted by CX\. So, Cf = CX ∪f Y , and SX = CX/X × {1}\. \(d\) The join X ∗ (Budyak 30) - ∗ Y of the spaces X, Y is deﬁned to be the double mapping cylinder of the diagram p1 p2 X ←− X × Y −→ Y\. For instance, X ∗ S 0 = SX (Budyak 30) - fn−1 fn fn+1 3\.19\. Deﬁnition\. Given a sequence X = {· · · −−−→ Xn −→ Xn+1 −−−→ · · · } of maps, deﬁne its telescope T X to be the space T X := \(Xn × [n, n + 1]\) ∼, where under ∼, \(x, n + 1\) ∈ Xn × [n, n + 1] is identiﬁed with \(fn \(x\), n + 1\) ∈ X n+1 × [n + 1, n + 2]\. (Budyak 30) - 3\.20\. Deﬁnition\. Let {\(Xi , xi \)} be a family of pointed spaces\. \(a\) The pointed direct product is the pointed space \(X i , xi \) := Xi , ∗ where ∗ is the point {xi }\. \(b\) The wedge is the pointed space i Xi \(X Xi i , xi \) := i ,∗ \. ∪i {xi } i where ∗ is the image of ∪i {xi }\. (Budyak 31) - \(c\) The obvious injective maps \(Xi , xi \) → Xi , ∗ yield an injective map \(∨i Xi , ∗\) → Xi , ∗ \. Generally speaking, this map is not closed, but it is closed for a ﬁnite set of spaces\. So, given two pointed spaces \(X, ∗\), \(Y, ∗\), we deﬁne the smash product \(X, ∗\) × \(Y, ∗\) \(X, ∗\) × \(Y, ∗\) \(X, ∗\) ∧ \(Y, ∗\) := \. \(X, ∗\) ∨ \(Y, ∗\) (Budyak 31) - 3\.21\. Deﬁnition\. Let {\(Xi , xi \)} be a family of copies of a pointed space \(X, x\)\. We deﬁne the folding map π : ∨\(Xi , xi \) → X, to be the unique map π such that π|Xi = 1X \. (Budyak 31) - 3\.23\. Deﬁnition\. \(a\) The reduced mapping cylinder of a pointed map f : \(X, ∗\) → \(Y, ∗\) is the space M f = \(X × [0, 1] ∪f Y \)/\(∗ × [0, 1]\)\. Note that the base points of X and Y yield the same point ∗ ∈ M f ; we agree that ∗ is the base point of M f \. (Budyak 31) - \(b\) The reduced mapping cone of f is deﬁned to be Cf = M f /\(X × {1}\)\. It is a pointed space in the obvious way: its base point is the image of the base point of M f \. (Budyak 32) - \(c\) The reduced mapping cone of the constant map \(X, ∗\) → \(pt, ∗\) is called the reduced suspension over a space X and denoted by SX\. Furthermore, we can deﬁne the iterated reduced suspension S n X, and S n turns out to be a functor on W • , see 3\.18\(e\)\. (Budyak 32) - \(d\) The reduced telescope of a sequence f n−1 fn fn+1 fn−1 fn fn+1 X = {· · · −−−→ Xn −→ Xn+1 −−−→ · · · } of pointed maps is deﬁned to be the pointed space T X := \(Xn × [n, n + 1]\) ∼, where \(x, n + 1\) ∈ Xn × [n, n + 1] is identiﬁed with \(fn \(x\), n + 1\) ∈ Xn+1 × [n + 1, n + 2] and all the points of the form \(∗, t\) are identiﬁed\. (Budyak 32) - \(In fact, the reduced and unreduced cone \(cylinder, etc\.\) of any map\(s\) of CW -spaces are homotopy equivalent (Budyak 32) - Prove as an exercise that SX ∼ = S 1 ∧ X for every X ∈ W • \. (Budyak 32) - 3\.24\. Deﬁnition\. Given a pair \(X, A\), the inclusion i : A → X is called a coﬁbration if it satisﬁes the homotopy extension property, i\.e\., given maps g : X → Y and F : A × I → Y such that F |A × {0} = g|A, there is a map G : X × I → Y such that G|X × {0} = g and G|A × I = F \. In this case we also say that \(X, A\) is a coﬁbered pair\. (Budyak 32) - 3\.25\. Proposition\. \(i\) \(X, A\) is a coﬁbered pair iﬀ every map h : X × {0} ∪ A × I → Y can be extended to a map X × I → Y \. \(ii\) \(X, A\) is a coﬁbered pair iﬀ X × {0} ∪ A × I is a retract of X × I (Budyak 33) - 3\.26\. Proposition\. \(i\) For every map f : X → Y , the inclusion i : X = X × {1} → M f is a coﬁbration\. In particular, every map is homotopy equivalent to a coﬁbration\. \(ii\) Let \(X, A\) be a coﬁbered pair\. Then Ci X/A\. \(iii\) Let \(X, A\) be a coﬁbered pair\. If A is contractible then the collapsing map c : X → X/A is a homotopy equivalence\. (Budyak 33) - 3\.28\. Deﬁnition\. A pointed space \(X, x0 \) is called well-pointed if \(X, {x0 }\) is a coﬁbered pair\. (Budyak 34) - 3\.29\. Lemma \(Puppe [1]\)\. Let f : \(X, x0 \) → \(Y, y0 \) be a pointed map of well-pointed spaces\. If f : X → Y is a homotopy equivalence then f : \(X, x0 \) → \(Y, y0 \) is a pointed homotopy equivalence\. (Budyak 34) - \(c\) A long coﬁber sequence is a sequence \(ﬁnite or not\) ··· − → Xi − → Xi+1 − → Xi+2 − → ··· where every pair of adjacent morphisms forms a coﬁber sequence\. (Budyak 36) - u v 3\.38\. Deﬁnition\. \(a\) A strict coﬁber sequence is a diagram A − →B − →C where u : A → B is a map and v is the canonical inclusion as in \(3\.17\)\. f g \(b\) A sequence X − →Y − → Z is called a coﬁber sequence if there exists a homotopy commutative diagram f g X −−−−→ Y −−−−→ Z ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ c ⏐ a b c u v A −−−−→ B −−−−→ C such that all the vertical arrows are homotopy equivalences and the bottom row is a strict coﬁber sequenc (Budyak 36) - 3\.39\. Proposition\. Let f : X → Y be an arbitrary map of pointed spaces, and let g : Y → Cf = Z be the canonical inclusion\. Then C\(g\) SX = S 1 ∧ X\. Moreover, there is a long coﬁber sequence n f Sn f g Sf Sg Sn f Sn g X− →Y − →Z− → SX −−→ SY −→ · · · − → S n X −−→ S n Y −−→ · · · \. (Budyak 37) - Proposition 3\.39 was originally proved by Puppe [1]\. Because of this, the long coﬁber sequence is often refered to as the Puppe sequence (Budyak 37) - Recall that a map f : X → Y of CW -complexes is called cellular if f \(X \(n\) \) ⊂ Y \(n\) for every n\. (Budyak 37) - 3\.41\. Theorem\. \(i\) Let i : X \(n\) → X be the inclusion\. Then i∗ : π i \(X \(n\) , ∗\) → πi \(X, ∗\) is an isomorphism for i < n and an epimorphism for i = n\. \(ii\) Every map f : X → Y of CW -complexes is homotopic to a cellular map\. (Budyak 37) - We recall that i : A → X is a coﬁbration for every CW -pair \(X, A\)\. In particular, every pointed CW -space is well-pointed, and so we can safely omit base points from notation\. (Budyak 37) - Every CW -space is Hausdorﬀ \(and so weak Hausdorﬀ\) and compactly generated\. Thus, when we talk about products \(or smash products\) of CW complexes we follow 3\.6\. Then the direct product X × Y and the smash product X ∧ Y of two CW -spaces X, Y are also CW -spaces\. Note that, for every cellular map f : X → Y , the spaces M f and Cf are CW -complexes in an obvious canonical way, see e\.g\. Fritsch–Piccinini [1]\. In particular, the suspension SX of a CW -complex X is a CW -complex\. (Budyak 38) - \(b\) A map f : X → Y is called a Whitehead equivalence if f∗ : π n \(X, x0 \) → πn \(Y, f \(x0 \)\) is a bijection for every n ≥ 0 and every x0 ∈ X\. (Budyak 38) - We say that two spaces X, Y are CW -equivalent if there is a sequence a 0 a1 ai−1 ai an−1 a0 a1 ai X = X0 −−−− X1 −−−− · · · −−−−− − Xi −−−− · · · −−−−− − Xn = Y where every ai is a Whitehead equivalence (Budyak 38) - 3\.43\. Remark\. Traditionally, CW -equivalences, as well as Whitehead equivalences, are called weak equivalences\. We refrain from using this terminology in this book because these names are not quite compatible with the concept of weak homotopy (Budyak 38) - 3\.44\. Proposition–Deﬁnition\. For every topological space X, there is a Whitehead equivalence f : Y → X where Y is a CW -space\. Every such CW space Y is called a CW -substitute for X\. (Budyak 38) - 3\.45\. Lemma\. Let h : Y → Z be a Whitehead equivalence\. \(i\) Let \(X, A\) be a CW -pair, and let f : A → Y, u : X → Z be two maps such that hf = u|A\. Then there is a map g : X → Y such that g|A = f and hg u\. (Budyak 39) - Proof\. \(i\) This is an exercise in elementary obstruction theory, see e\.g\. Switzer [1], 6\.30\. (Budyak 39) - 3\.47\. Corollary \(the Whitehead Theorem\)\. If h : Y → Z is a Whitehead equivalence of CW -spaces then h is a homotopy equivalence\. (Budyak 40) - 1\.1\. Deﬁnition\. \(a\) A spectrum E is a sequence {En , sn }, n ∈ Z, of CW complexes En and CW -embeddings sn : SEn → En+1 \(i\.e\., sn \(SEn \) is a subcomplex of En+1 \)\. (Budyak 42) - \(e\) For every CW -complex X the spectrum Σ∞ X is deﬁned as follows: pt, if n < 0, ∞ \(Σ∞ X\)n = S n X, if n ≥ 0, and sn : S\(S n X\) → S n+1 X, for n ≥ 0, are the identity maps\. For example, the spectrum Σ∞ S 0 is the sphere spectrum {S n , in }, where i n : SS = n −→ S n+1 \. (Budyak 42) - 1\.2\. Deﬁnition\. \(a\) A cell of a spectrum E is a sequence {e, Se, \. \. \. , S k e, \. \. \. } where e is a cell of any En such that e is not the suspension of any cell of E n−1 \. If e is a cell of En of dimension d then the dimension of the cell {e, Se, \. \. \. , S k e, \. \. \. } of E is d − n\. Furthermore, the base points of En ’s yield the cell of dimension −∞\. (Budyak 43) - \(b\) A subspectrum F of a spectrum E is coﬁnal \(in E\) if every cell of E is eventually in F , i\.e\., for every cell e of En there exists m such that S m e belongs to Fn+m \. (Budyak 43) - \(g\) A suspension spectrum is a spectrum of the form Σk Σ∞ X where X is a pointed space and k ∈ Z\. (Budyak 43) - 1\.3\. Deﬁnition\. \(a\) Let E = {En , sn } and F = {Fn , tn } be two spectra\. A map f from E to F \(i\.e\., a map f : E → F \) is a family of pointed cellular maps fn : En → Fn such that fn+1 sn = tn ◦Sfn for all n\. (Budyak 43) - Let S = Σ∞ S 0 be the spectrum of spheres\. The group [Σk S, E] is called the k-th homotopy group of E and denoted by πk \(E\)\. It is easy to see that π k \(E\) = lim πk+N \(EN \) where the direct limit is that of the direct system N →∞ \(sN \)∗ · · · → πk+N \(EN \) → πk+N +1 \(SEN \) −−−→ πk+N +1 \(EN +1 \) → · · · , (Budyak 45) - 1\.9\. Deﬁnition\. \(a\) Two maps g0 , g1 : E → F of spectra are called homotopic if there exists a map G : E ∧ I + → F \(called a homotopy\) such that G coincides with gi on the subspectrum E ∧ {i, ∗}, i = 0, 1, of E\. In this case we write g0 g1 or G : g0 g1 \. (Budyak 45) - \(b\) Two morphisms ϕ0 , ϕ1 : E → F of spectra are called homotopic, if there exists a coﬁnal subspectrum E of E and two maps gi : E → F, gi ∈ ϕ i , i = 0, 1, such that g0 |E g1 |E \. It is straightforward to show that homotopic morphisms form equivalence classes, and in particular we can deﬁne the homotopy class [ϕ] of a morphism ϕ to be the set of all morphisms homotopic to ϕ\. The set of all homotopy classes of morphisms E → F is denoted by [E, F ]\. (Budyak 45) - Thus, we can deﬁne a category H S with spectra as objects and sets [E, F ] as sets of morphisms\. Isomorphisms of H S are called equivalences \(of spectra\), and we use the notation E F when E is equivalent to (Budyak 45) - o F \. (Budyak 45) - In particular, if E = Σ∞ X then π k \(E\) is just the stable homotopy group Πk \(X\) \(denoted also by πkst \(X\)\) (Budyak 46) - An analog of the Whitehead Theorem is valid for spectra\. 1\.10\. Theorem\. A morphism ϕ : E → F is an equivalence iﬀ the induced homomorphism ϕ∗ : πk \(E\) → πk \(F \) is an isomorphism for every integer k\. (Budyak 46) - One of the important advantages of the category H S is that the suspension operator is invertible there\. 1\.11\. Proposition\. The spectra S 1 ∧ E and ΣE are equivalent\. (Budyak 46) - 1\.12\. Deﬁnition\. \(a\) A strict coﬁber sequence of spectra is a diagram ϕ ψ E− →F − → Cϕ where ϕ : E → F is a morphism of spectra \(resp\. map of spaces\) and ψ is a canonical inclusion as in 1\.7\(b\)\. (Budyak 46) - 1\.14\. Proposition\. For every spectrum F the function {i∗ λ } : [∨λ E\(λ\), F ] → [E\(λ\), F ], where {i∗λ }\(f \) = {f iλ }, λ is a bijection\. (Budyak 47) - f g 1\.13\. Lemma\. \(i\) If X − → Y − → Z is a coﬁber sequence of spectra, then f g h there exists a map h : Z → ΣX such that X − →Y − →Z − → ΣX is a long coﬁber sequence\. (Budyak 47) - Since [E, F ] = [Σ2 E, Σ2 F ] = [S 2 ∧ E, Σ2 F ] \(the last equality follows from 1\.11\), [E, F ] admits a natural structure of an abelian group (Budyak 48) - Thus, H S is an additive category\. (Budyak 48) - f g In view of 1\.11 and 1\.13\(i\), every coﬁber sequence X − →Y − → Z yields a long coﬁber sequence −1 \. \. \. → Σ −1 Y Σ−1 g f g Σf −−−→ Σ−1 Z − →X − →Y − → Z → ΣX −−→ ΣY − → ··· \. (Budyak 48) - −1 \. \. \. → Σ −1 Y Σ−1 g f g Σf −−−→ Σ−1 Z − →X− →Y − → Z → ΣX −−→ ΣY − → ··· yields the exact sequences ∗ g∗ f∗ g∗ − [Σ−1 Z, E] ← ··· ← − [X, E] ←− [Y, E] ←− [Z, E] ← − [ΣX, E] ← − ··· → [E, Σ ··· − −1 Z] f∗ g∗ − → [E, X] −→ [E, Y ] −→ [E, Z] − → [E, ΣX] − → ··· of abelian groups and homomorphisms (Budyak 48) - The ﬁrst of the above sequences is similar to a sequence which holds for a coﬁbration X → Y with coﬁber Z , while the second one is similar to a sequence which holds for a ﬁbration X → Y with ﬁber Σ−1 Z \. Thus, the diﬀerence between ﬁbrations and coﬁbrations disappears in the category H S \. For this reason we call Σ −1 Cϕ the fiber of a morphism ϕ\. (Budyak 48) - ular, π∗ \(∨λ E\(λ\)\) ∼ = ∼ = ⊕λ π∗ \(E\(λ\)\) (Budyak 49) - Hence, the element 1X ⊕ 1X of the right hand side yields a \(unique up to homotopy\) morphism ∇ : X → X ∨ X\. We leave it to the reader to show that addition in [X, E] is given by the composition ∼ = ∇∗ ∼ = ∇∗ [X, E] ⊕ [X, E] −→ [X ∨ X, E] −−→ [X, E]\. Because of this, we call ∇ coaddition (Budyak 49) - f g 1\.17\. Proposition\. Let X − →Y − → Z be a coﬁber sequence of spectra\. The following two conditions are equivalent: \(i\) The morphism g is inessential; \(ii\) There is a morphism s : Y → X such that f s 1Y \. (Budyak 49) - Furthermore, if these conditions hold then X Σ−1 Z ∨ Y \. (Budyak 50) - 1\.18\. Deﬁnition\. A prespectrum is a family {Xn , tn }, n ∈ Z, of pointed spaces Xn and pointed maps tn : SXn → Xn+1 \. A CW -prespectrum is a prespectrum {Xn , tn } such that every Xn is a CW -complex and every tn is a cellular map\. 1\.19\. Lemma–Deﬁnition\. For every prespectrum {Xn , tn }, there exist a spectrum E = {En , sn } and pointed homotopy equivalences fn : En → Xn such that the diagram SE Sfn n −−−−→ SXn ⏐ ⏐ ⏐ s ⏐ ⏐ ⏐ ⏐t n n E fn+1 n+1 −−−−→ Xn+1 commutes\. Every such spectrum E is called a spectral substitute of the prespectrum X (Budyak 50) - 1\.20\. Deﬁnition\. A prespectrum X = {Xn , tn } is called an Ω-prespectrum if for every n the map τn : Xn → ΩXn+1 adjoint to tn is a homotopy equivalence\. A spectrum is called an Ω-spectrum if it is an Ω-prespectrum\. 1\.21\. Proposition \(cf\. Adams [5]\)\. Every spectrum E = {En , sn } is equivalent to some Ω-spectrum\. (Budyak 51) - Some authors have developed a ﬁner theory by indexing terms of a spectrum not by integers but by ﬁnite dimensional subspaces of R∞ \. This approach was suggested by Puppe [2] and May [3]\. Such spectra are very useful for working with some ﬁne geometry\. However, the foundations of this theory are quite complicated\. For our purposes, the mass of preliminaries outweighs the gain; thus we do not use these spectra here and so do not dwell on them\. However, they seem to be very useful for advanced homotopy theory (Budyak 54) - One can introduce a smash product E ∧ F of spectra E, F as a generalization of the smash product E ∧ X of a spectrum and a space\. (Budyak 54) - 2\.3\. Deﬁnition\. \(a\) A morphism u : S → A ∧ A⊥ is called a duality morphism, or simply a duality, between spectra A and A⊥ if for every spectrum E the homomorphisms u E : [A, E] → [S, E ∧ A ⊥ ], uE \(ϕ\) = \(ϕ ∧ 1A⊥ \)u and u E : [A⊥ , E] → [S, A ∧ E], uE \(ϕ\) = \(1A ∧ ϕ\)u are isomorphisms\. \(b\) A spectrum A⊥ is called dual to a spectrum A if there exists a duality S → A ∧ A⊥ \. By 2\.1\(ii\), in this case A is dual to A⊥ \. So, “to be dual” is a symmetric relation\. (Budyak 56) - 2\.6\. Remarks\. \(a\) Some authors deﬁne duality to be a morphism v : A ∧ A ⊥ → S such that F vE : [E, A⊥ ∧ F ] → [A ∧ E, F ] and F v E : [E, F ∧ A] → [E ∧ A⊥ , F ] are isomorphisms (Budyak 58) - 2\.9\. Corollary\. \(i\) Every ﬁnite CW -space X admits an n-dual ﬁnite CW space X for n large enough\. \(ii\) Every ﬁnite spectrum A admits a dual ﬁnite spectrum A⊥ \. (Budyak 60) - f g 2\.10\. Proposition\. If A − →B− → C is a coﬁber sequence of ﬁnite spectra, ⊥ ⊥ f ⊥ ⊥ g f then C ⊥ −−→ B ⊥ −−→ A⊥ is a coﬁber sequence\. (Budyak 60) - 2\.12\. Deﬁnition\. \(a\) A ring spectrum is a triple \(E, μ, ι\) where E is a spectrum and μ : E ∧ E → E \(the multiplication\) and ι : S → E \(the unit morphism, or the unit\) are certain morphisms with the following properties: (Budyak 60) - 2\.13\. Deﬁnition\. \(a\) A module spectrum over a ring spectrum \(E, μ, ι\), or an E-module spectrum, is a pair \(F, m\) where F is a spectrum and m : E ∧ F → F is a morphism such that the following diagrams commute up to homotopy: (Budyak 61) - it was noticed that many useful constructions of algebraic topology \(K-functor, \(co\)bordism, etc\.\) are formally similar to \(co\)homology theories\. Afterwards the reason for this phenomenon was clariﬁed: namely, most of these constructions satisfy all the Eilenberg–Steenrod axioms except the so-called dimension axiom\. So, it seemed reasonable to consider the objects satisfying these axioms\. These objects were called extraordinary \(co\)homology theories\. However, later mathematicians came to call these objects just \(co\)homology theories, while H\(−; G\) got the name ordinary \(co\)homology theory\. 5 Now this terminology is commonly accepted, and we use it in this book\. (Budyak 62) - It is easy to see that we have a category of homology theories and their morphisms\. In particular, the equivalence \(isomorphism\) of homology theories is deﬁned in the usual way: it is a morphism of homology theories which is also a natural equivalence of functors\. (Budyak 63) - The exact sequences as in 3\.1\(2\), 3\.2\(iv\), and 3\.2\(v\) are known as the exact sequence of a pair, the exact sequence of a triple, and the Mayer– Vietoris exact sequence (Budyak 65) - The family {hn , sn } is called a reduced homology theory \(on K \) corresponding • to {hn , ∂n }\. (Budyak 65) - This proposition shows that there is a bijective correspondence between unreduced and reduced homology theories\. In other words, every unreduced homology theory is completely determined by its reduced form (Budyak 68) - The groups hi \(pt, ∅\) = hi \(S hi \(S 0 , ∗\) are called the coeﬃcient groups of the homology theory {hn , ∂n }\. To justify this term, note that H∗ \(pt, ∅; A\) = A for every abelian group A\. (Budyak 68) - It is possible and useful to introduce \(co\)homology theories on spectra\. Consider the following full subcategories of S : Sfd : its objects are all ﬁnite dimensional spectra; Ss : its objects are all suspension spectra; Ssfd : its objects are all spectra of the form Σn Σ∞ X, n ∈ Z, X ∈ Cfd • ; Sf : its objects are all ﬁnite spectra\. 3\.10\. Deﬁnition\. Let L be one of the categories S , Sfd , Ss , Ssfd , Sf , and let Σ : L → L be the functor deﬁned in 1\.1\(d\)\. \(a\) A homology theory on L is a family {hn , sn }, n ∈ Z of covariant functors hn : L → A G and natural transformations sn : hn → hn+1 Σ satisfying the following axioms: (Budyak 70) - 3\.12\. Proposition\. Let L be as in 3\.10, and let {hn , ŝn } be a cohomology theory on L \. Then: \(i\) hn \(X ∨ Y \) ∼ = ∼ = hn \(X\) ⊕ hn \(Y \)\. f g \(ii\) For every coﬁber sequence X − →Y − → Z of spectra there is a natural exact sequence ∗ f∗ g∗ f∗ → hn−1 \(X\) − ··· − → hn \(Z\) −→ hn \(Y \) −→ hn \(X\) − → hn+1 \(Z\) − → ··· \. \(iii\) Let A, B be two subspectra of a spectrum X\. Set C = A ∩ B\. Then there is a natural \(Mayer-Vietoris\) exact sequence ··· − → hn \(X\) − → hn \(A\) ⊕ hn \(B\) − → hn \(C\) − → hn−1 \(X\) − → ··· \. (Budyak 72) - 3\.22\. Construction\. Let E be an arbitrary spectrum\. \(a\) Deﬁne covariant functors En : S → A G where En \(X\) := πn \(E ∧ X\) for every X ∈ S and En \(f \) := πn \(1E ∧ f \) for every morphism f : X → Y of spectra\. Furthermore, deﬁne sn : En → En+1 Σ to be the composition En \(X\) = πn \(E ∧ X\) = πn+1 \(Σ\(E ∧ X\)\) πn+1 \(E ∧ ΣX\) = En+1 \(ΣX\) for every X ∈ S \. By 1\.15 and 2\.1\(vi\), \(En , sn \) is a homology theory on S , and, by 1\.16\(iii\) and 2\.1\(v\), it is additive\. (Budyak 78) - hus, every spectrum yields a \(co\)homology theory on S (Budyak 78) - \. So, we have a functor from spectra to \(co\)homology theories\. In particular, equivalent spectra yield isomorphic \(co\)homology theories\. According to 3\.22, one can assign a \(co\)homology theory to a spectrum\. This situation turns out to be invertible, see Ch\. III, §3 below\. ∗ 3\.23\. Proposition\. For every spectrum E, the cohomology theory E ∗ is dual to the homology theory E∗ \. (Budyak 78) - 3\.25\. Proposition\. If E = {En } is an Ω-spectrum, then for every space X i \(X\) ∼ there is a natural equivalence E = [X, Ei ]• \. (Budyak 79) - 3\.26\. Corollary\. For every spectrum E and every i, the functor i : E H C • → A G , X → E i \(X\), E is representable\. (Budyak 80) - \.27\. Proposition-Deﬁnition\. Given a spectrum E, let Ω∞ E denote a 0 : representing space for E H C • → A G \(i\.e\., Ω∞ E = F0 for some Ωspectrum F equivalent to E\)\. This space Ω∞ E is called the inﬁnite delooping of E and has the following properties: (Budyak 80) - this section we develop the homotopy theory of spectra\. Namely, we discuss Postnikov towers, Cartan killing constructions, Serre theory of classes of abelian group, etc\., for spectra\. \(We assume that the reader knows these notions in the case of spaces; otherwise he can ﬁnd them e\.g\. in Mosher– Tangora [1]\.\) Closely related material is exposed in Dold [3] and Margolis [1]\. (Budyak 88) - 4\.9\. Theorem \(the Universal Coeﬃcient Theorem\)\. For every spectrum E and every abelian group G, there are exact sequences 0 → Ext\(Hn−1 \(E\), G\) → H n \(E; G\) → Hom\(Hn \(E\), G\) → 0 and 0 → Hn \(E\) ⊗ G → Hn \(E; G\) → Tor\(Hn−1 \(E\), G\) → 0\. In particular, H0 \(H\(A\); B\) ∼ = A ⊗ B, H 0 \(H\(A\); B\) ∼ = ∼ = Hom\(A, B\)\. (Budyak 91) - 4\.11\. Theorem \(the Künneth Theorem\)\. Let k be a ﬁeld, and let E, F be a pair of spectra\. \(i\) The homomorphism μ E,F : H∗ \(E; k\) ⊗ H∗ \(F ; k\) → H∗ \(E ∧ F ; k\) is an isomorphism\. In particular, H n \(E ∧ F ; k\) ∼ = ∼ = Hi \(E; k\) ⊗k Hj \(F ; k\)\. i+j=n \(ii\) Assume that E is bounded below and F has ﬁnite type\. Then the homomorphism μ E,F : H ∗ \(E; k\) ⊗ H ∗ \(F ; k\) → H ∗ \(E ∧ F ; k\) is an isomorphism\. In particular, H n \(E ∧ F ; k\) ∼ = ∼ = H i \(E; k\) ⊗k H j \(F ; k\)\. i+j=n (Budyak 93) - 4\.12\. Deﬁnition\. A Postnikov tower of a spectrum E is a homotopy commutative diagram of spectra (Budyak 94) - 4\.13\. Theorem\. Every spectrum E has a Postnikov tower (Budyak 94) - 4\.19\. Deﬁnition\. The element κn ∈ H n+1 \(E\(n−1\) ; πn \(E\)\) is called the n-th Postnikov invariant of E\. (Budyak 97) - Now we apply the Serre class theory \(see Serre [1], Mosher–Tangora [1]\) to spectra\. C (Budyak 98) - 4\.21\. Deﬁnition\. \(a\) A Serre class is a family of abelian groups C satisfying the following axiom: If 0 → A → A → A → 0 is a short exact sequence, then A is in C iﬀ both A and A are in C\. (Budyak 98) - C iﬀ both A and A are in C\. \(b\) Let H\(A\) denote the Eilenberg–Mac Lane spectrum of an abelian group A\. A Serre class is called stable if it satisﬁes the following axiom: If A ∈ C, then Hi \(H\(A\)\) ∈ C for every i\. (Budyak 98) - 4\.23\. Proposition\. \(i\) The class of all ﬁnite abelian groups is a stable Serre class\. \(ii\) The class of all ﬁnitely generated abelian groups is a stable Serre class\. (Budyak 98) - \(iii\) Given a prime p, let C be the class of all abelian groups having pprimary exponents \(i\.e\., for every A ∈ C there exists k such that pk A = 0\)\. Then C is a stable Serre class\. \(iv\) Given a prime p, the class of all ﬁnite p-primary abelian groups is a stable Serre class\. (Budyak 99) - 4\.32\. Theorem-Deﬁnition\. For every abelian group A, there exists a spectrum M \(A\) with the following properties: \(i\) πi \(M \(A\)\) = 0 for i < 0; \(ii\) π0 \(M \(A\)\) = A = H0 \(M \(A\)\); \(iii\) Hi \(M \(A\)\) = 0 for i = 0\. (Budyak 104) - Moreover, these properties determine M \(A\) uniquely up to equivalence\. This spectrum M \(A\) is called the Moore spectrum of the abelian grou (Budyak 104) - Let Q be the ﬁeld of rational numbers\. Let p be a prime, and let Z[p] be the subring of Q consisting of all irreducible fractions with denominators relatively prime to p\. The Z[p]-localization of an abelian group A is the homomorphism A → A ⊗ Z[p], a → a ⊗ 1\. The group A ⊗ Z[p] is simpler than A in a certain sense: for example, it has no q-torsion if \(p, q\) = 1\. On the other hand, if we know the groups A ⊗ Z[p] for all p then we can obtain a lot of information about A; for example, if A is ﬁnitely generated then it is completely determined by the groups A ⊗ Z[p], where p runs through all primes\. So, we can describe an abelian group A via descriptions of the simpler groups A ⊗ Z[p], and this trick is very eﬀective\. For example, it is ∗ very convenient to describe the ring H ∗ \(HZ\) of cohomology operations via the rings H ∗ \(HZ[p]; Z[p]\)\. Also, localization enables us to ignore the torsions which are irrelevant to a particular problem\. (Budyak 106) - More generally, it makes sense to consider subrings Λ of Q\. In this case the localization A → A ⊗ Λ deletes the q-torsion with q ∈ S, where S is the set of denominators of all irreducible fractions of Λ\. (Budyak 106) - It is remarkable that the localization can be transferred from algebra to topology, and, in particular, one can consider the Z[p]-homotopy type of a space and a spectrum (Budyak 106) - Let Λ be a subring of Q ; its additive group is also denoted by Λ\. Let π be an abelian group\. π 5\.1\. Deﬁnition\. The homomorphism l = lΛ : π → π ⊗ Λ, a → a ⊗ 1 is called the Λ-localization of π\. (Budyak 106) - f g 5\.3\. Proposition\. If E − → F − → G is a coﬁber sequence of spectra then E fΛ gΛ Λ −→ FΛ −→ GΛ is\. In particular, C\(fΛ \) = \(Cf \)Λ for every morphism f : E → F of spectra\. (Budyak 107) - 5\.5\. Corollary\. There are natural isomorphisms π i \(EΛ \) ∼ = πi \(E\) ⊗ Λ, Hi \(EΛ \) ∼ = Hi \(E\) ⊗ Λ j∗ such that the homomorphisms πi \(E\) −→ πi \(EΛ \) ∼ πi \(E\) = ⊗ Λ, Hi \(E\) −j→ ∗ H i \(EΛ \) ∼ = ∼ = Hi \(E\)⊗Λ have the form x → x⊗1\. So, j Λ-localizes homotopy and homology groups\. In particular, every Λ-local spectrum has Λ-local homotopy and homology groups\. (Budyak 108) - 5\.7\. Lemma\. Let E be an arbitrary spectrum, and let Cj be the cone of the localization j : E → EΛ \. Then H ∗ \(Cj; π\) = 0 for every Λ-local group π\. (Budyak 110) - This proposition enables us to construct localizations via Postnikov towers\. (Budyak 112) - This approach enables us to construct the localization of spaces also\. The main results of this theory are Theorems 5\.12 and 5\.13 below (Budyak 112) - 5\.12\. Theorem–Deﬁnition\. For every simple space X there exist a simple space XΛ and a map j = jΛX : X → XΛ such that the homomorphisms π i \(X\) −→ πi \(XΛ \) ∼ = j∗ ∼ = π i \(X\) ⊗ Λ and Hi \(X\) −→ Hi \(XΛ \) j∗ ∼ = Hi \(X\) ⊗ Λ have the form x → x ⊗ 1\. So, j localizes homotopy and homology groups\. Every such a map j is called localization of X\. (Budyak 112) - As in 5\.2, a simple space X is called Λ-local if j : X → XΛ is a homotopy equivalence\. (Budyak 112) - 5\.13\. Theorem–Deﬁnition\. For every two simple spaces X, Y the following conditions are equivalent: \(i\) The map f : X → Y Λ-localizes homotopy groups; \(ii\) The map f : X → Y Λ-localizes homology groups; \(iii\) For every Λ-local space Z the map f ∗ : [Y, Z] → [X, Z] is a bijection\. If some \(and hence all\) of these conditions hold then there exists a homotopy equivalence h : XΛ → Y with f = hjΛX \. Furthermore, in this case we say that f localizes X\. (Budyak 113) - 5\.14\. Lemma\. For every two spectra E, F there exists an equivalence ϕ : \(E ∧ F \)Λ → EΛ ∧ FΛ such that the diagram E∧F j E∧F E∧F −−−−→ \(E ∧ F \)Λ ⏐ ⏐ ⏐ ⏐ j E ∧j F ⏐ ⏐ ⏐ ϕ ⏐ ∧j E Λ ∧ FΛ EΛ ∧ FΛ is homotopy commutative\. (Budyak 113) - 6\.1\. Deﬁnition\. \(a\) An algebra over R \(or simply an R-algebra\) is a triple \(A, μ, η\), where A is an R-module and μ : A ⊗ A → A, η : R → A are R-homomorphisms such that the diagrams μ⊗1 η⊗1 1⊗η A ⊗ A ⊗ A −−−−→ A ⊗ A R ⊗ A −−−−→ A ⊗ A ←−−−− A ⊗ R ⏐ ⏐ ⏐ ⏐ ⏐ 1⊗μ ⏐ ⏐ ⏐ ⏐ ⏐μ ⏐ ⏐μ ⏐∼ =⏐ ∼ ⏐= ⏐ ⏐μ ⏐∼ =⏐ ∼ ⏐= μ A⊗A −−−−→ A A A A ∼ commute\. Here = denotes the canonical isomorphisms R ∼ ⊗ A = ∼ ∼ = A = A ⊗ R, e\.g\., A ∼ = A ⊗ R has the form a → a ⊗ 1\. Furthermore, μ is called the multiplication and η is called the unit homomorphism\. An algebra \(A, μ, η\) is commutative if μTA,A = μ\. (Budyak 116) - \(b\) A \(left\) module over an R-algebra \(A, μ, η\) is a pair \(M, ϕ\), where M is an R-module and ϕ : A ⊗ M → M is an R-homomorphism such that the following diagrams commute: μ⊗1 A ⊗ A ⊗ M −−−−→ A ⊗ M ⏐ ⏐ ⏐ 1⊗ϕ ⏐ ⏐ ⏐ ⏐ϕ ϕ A⊗M −−−−→ M ϕ A ⊗ M −−−−→ M ⏐ ⏐ ⏐∼ η⊗1⏐ ⏐= R⊗M R ⊗ M\. As usual, we shall simply say “algebra A” or “A-module M ”, omitting μ, η, ϕ, and we shall write ab instead of μ\(a ⊗ b\) and am instead of ϕ\(a ⊗ m\) (Budyak 116) - ote that every ring is a Z-algebra\. (Budyak 116) - 6\.2\. Deﬁnition\. A homomorphism f : A → B of R-algebras is an Rhomomorphism such that the ﬁrst two of the three diagrams below commute\. (Budyak 116) - A homomorphism h : M → N of A-modules is an R-homomorphism such that the third diagram commutes\. (Budyak 117) - 6\.4\. Deﬁnition\. An augmented R-algebra is a quadruple \(A, μ, η, ε\), where \(A, μ, η\) is an R-algebra and ε : A → R is a homomorphism of R-algebras \(called the augmentation\)\. An augmented algebra \(A, μ, η, ε\) is called connected if Ai = 0 for i < 0 and ε|A0 : A0 → R is an isomorphism\. (Budyak 117) - 6\.5\. Deﬁnition\. Let A be a connected R-algebra, and let Ā be the Rsubmodule consisting of all elements of positive degrees\. The ideal ĀĀ is denoted by Dec A, and its elements are called decomposable elements of A\. Given an A-module M , let GM denote the factor module M/ĀM \. Furthermore, GĀ := Ā/ Dec A usually is denoted by QA and is called \(not very aptly\) the set of indecomposable elements of A\. (Budyak 117) - 6\.7\. Deﬁnition\. \(a\) A coalgebra over R is a triple \(C, Δ, ε\), where C is an R-module and Δ : C → C ⊗ C, ε : C → R are R-homomorphisms such that the diagrams Δ C −−−−→ C ⊗C C C C ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Δ ⏐ ⏐ ⏐ ⏐ Δ⊗1 ∼ = ⏐ ⏐ ⏐ ⏐ ⏐ ⏐∼ Δ = 1⊗Δ ε⊗1 1⊗ε C ⊗ C −−−−→ C ⊗ C ⊗ C R ⊗ C ←−−−− C ⊗ C −−−−→ C ⊗ R commute\. Here Δ is called the comultiplication, or the diagonal map, or just the diagonal, and ε is called the augmentation, or the counit homomorphism\. A coalgebra \(C, Δ, ε\) is cocommutative if TC,C Δ = Δ\. (Budyak 117) - \(b\) A comodule over a coalgebra \(C, Δ, ε\) \(or, brieﬂy, a C-comodule\) is a pair \(V, ψ\), where V is a graded R-module and ψ = ψV is an Rhomomorphism V → C ⊗ (Budyak 118) - c\) A homomorphism h : C → D of R-coalgebras is an R-homomorphism such that ΔD ◦h = \(h ⊗ h\)◦ΔC and εD ◦h = εC \. A homomorphism f : V → W of comodules over a coalgebra \(C, Δ, ε\) is an R-homomorphism such that \(1 C ⊗ f \)◦ψV = ψW ◦f \. (Budyak 118) - The duality between algebras and coalgebras is exhibited not only in the deﬁning diagrams\. For instance, let R be a ﬁeld k\. Given a k-vector space C, ∗ consider the dual vector space C ∗ = Homk \(C, k\)\. If C is a k-algebra \(C, μ, η\) ∗ obtains a natural k-coalgebra structure \(C ∗ then C ∗ obtains a natural k-coalgebra structure \(C ∗ , Δ, ε\) provided that every component Cn of C is a ﬁnite dimensional k-vector space (Budyak 118) - Namely, Δ\(f \)\(a ⊗ b\) = f \(ab\), ε = η ∗ : C ∗ → k ∗ = k\. (Budyak 118) - 6\.8\. Lemma\. For every C-comodule \(V, ψ\), the function t : HomC \(V, C ⊗ M \) → HomR \(V, M \), t\(f \) = \(ε ⊗ 1\)f, is bijective\. (Budyak 118) - 6\.9\. Deﬁnition\. A coalgebra \(C, Δ, ε\) is called connected if Ci = 0 for i < 0 and ε|C0 : C0 → R is an isomorphism\. In this case the element v = ε−1 \(1\) is called the counit of C\. (Budyak 119) - 6\.11\. Examples\. \(a\) Let p be a prime, and let \(E, μ, ι\) be a ring spectrum of ﬁnite Z[p]-type\. Then H ∗ \(E; Z/p\) has the natural structure of a Z/pcoalgebra\. (Budyak 119) - \(b\) If the ring spectrum \(E, μ, ι\) in \(a\) is commutative then H ∗ \(E; Z/p\) is a cocommutative coalgebra\. (Budyak 119) - \(c\) If E is a spectrum as in \(a\) and F is any E-module spectrum of ﬁnite Z[p]-type, then H ∗ \(F ; Z/p\) admits a structure of a comodule over H ∗ \(E; Z/p\)\. (Budyak 119) - \(e\) For every CW -space X and every ﬁeld k we have a coalgebra H∗ \(X; k\) (Budyak 120) - ∗ \(f\) Dually to \(e\), H ∗ \(X; k\) is a k-algebra for every ﬁeld k and CW -space X of ﬁnite type\. (Budyak 120) - 6\.12\. Deﬁnition\. Let \(C, Δ, ε\) be a connected coalgebra with counit v\. \(a\) An element m ∈ C is called primitive if Δ\(m\) = m ⊗ v + v ⊗ m\. The set \(in fact, R-submodule\) of all primitive elements of C is denoted by Pr C\. \(b\) Let \(V, ψ\) be a C-comodule\. An element m ∈ V is called simple if ψ\(m\) = v ⊗ m\. The R-submodule of simple elements of V is denoted by Si V \. (Budyak 120) - 6\.13\. Remarks\. \(a\) Under the duality between algebras and coalgebras over a ﬁeld, Pr C is dual to QC ∗ \. \(b\) Sometimes simple elements are also called primitive, but we do not like this because of the danger of confusion: Pr C = Si C where C is regarded as a coalgebra on the left and as comodule on the right\. (Budyak 120) - 6\.14\. Lemma\. Let h : C → D be a homomorphism of connected coalgebras over a ﬁeld k\. If the map h| Pr C is injective in dimensions ≤ d then h is\. (Budyak 120) - 6\.15\. Construction\. \(Boardman [1]\)\. Given a connected coalgebra C, we deﬁne a ﬁltration Fm C by setting C n if n ≤ m, \(F m C\)n = 0 otherwise\. (Budyak 121) - m Fm V = V for every V \. \(ii\) F0 V = Si V \. \(iii\) f \(Fm V \) ⊂ Fm W for every C-comodule homomorphism f : V → W (Budyak 121) - 6\.18\. Deﬁnition\. Let M be a free R-module\. Given a coalgebra \(C, Δ, ε\), deﬁne its cofree M -extension to be the C-comodule \(V, ψ\) where V = C ⊗ M and ψ\(c ⊗ x\) = Δ\(c\) ⊗ x, c ∈ C, x ∈ M \. A C-comodule V is called cofree if there is M such that V is isomorphic to the cofree M -extension of C\. (Budyak 121) - 6\.19\. Lemma\. For every m, the homomorphism ψ : Fm V /Fm−1 V → C m ⊗ Si V is a monomorphism, and it is an isomorphism if V is a cofree C- (Budyak 122) - comodule (Budyak 122) - 6\.22\. Deﬁnition\. A Hopf algebra over R is a quintuple \(A, μ, η, Δ, ε\) such that \(A, μ, η, ε\) is an augmented R-algebra, \(A, Δ, ε\) is an R-coalgebra, Δ : A → A ⊗ A and ε : A → R are homomorphisms of algebras, and η : R → A is a homomorphism of coalgebras\. (Budyak 122) - Note that if A is a Hopf algebra over a ﬁeld k and if dimk An < ∞ for every n then A∗ = Hom\(A, k\) is a Hopf algebra also (Budyak 122) - 6\.24\. Deﬁnition\. Let A be a Hopf algebra over R\. \(a\) An A-module algebra is a quadruple \(M, μ, η, ϕ\), where \(M, μ, η\) is an R-algebra and \(M, ϕ\) is an A-module such that μ : M ⊗ M → M is a homomorphism of A-modules and ϕ : A ⊗ M → M is a homomorphism of R-algebras\. A homomorphism of A-module algebras is a homomorphism of A-modules which is at the same time a homomorphism of R-algebras\. \(b\) An A-comodule algebra is a quadruple \(M, μ, η, ψ\), where \(M, μ, η\) is an R-algebra and \(M, ψ\) is an A-comodule such that μ : M ⊗ M → M is a homomorphism of A-comodules and ψ : M → A ⊗ M is a homomorphism of R-algebras\. \(c\) An A-module coalgebra is a quadruple \(V, Δ, ε, ϕ\), where \(V, Δ, ε\) is an R-coalgebra and \(V, ϕ\) is an A-module such that Δ : V → V ⊗ V is a homomorphism of A-modules and ϕ : A ⊗ V → V is a homomorphism of R-coalgebras\. (Budyak 123) - 6\.25\. Recollection\. A very important example of a Hopf algebra over the ﬁeld Z/p, p prime, is the Steenrod algebra Ap = H ∗ \(HZ/p; Z/p\) = [HZ/p, Σd HZ/p]\. (Budyak 123) - All that we need to know about Ap can be found in Steenrod–Epstein [1], Margolis [1]\. (Budyak 124) - \. The Steenrod algebra A2 is generated by elements Sq i of dimension i, where i = 1, 2, \. \. \. , and all relations among Sq i follow from the Adem relations [a/2] [a/2] b−c−1 b = Sq a+b−c Sq c \(6\.26\) Sq a Sq b = Sq a+b−c Sq c for a < 2b, a − 2c c=0 k where Sq 0 := 1\. The comultiplication has the form Δ\(Sq k \) = Sq i ⊗Sq k−i \. (Budyak 124) - The Steenrod algebra Ap , p > 2, is generated by elements β \(the Bockstein homomorphism\) and P i , dim β = 1, dim P i = 2i\(p − 1\), i = 1, 2, \. \. \. (Budyak 124) - Note that the Hopf algebra Ap∗ can be described as H∗ \(H\), where H = HZ/p\. T (Budyak 125) - 27\. Examples\. \(a\) For every H-space X and every ﬁeld k, we have a Hopf algebra H∗ \(X; k\): t (Budyak 125) - Similarly, H ∗ \(X; k\) is a Hopf algebra for every H-space X of ﬁnite ∗ type, and H∗ \(X; k\) is the Hopf algebra dual to H ∗ \(X; k\)\. (Budyak 125) - \(b\) Let H\(−\) denote H\(−; Z/p\)\. For every spectrum E, the group H ∗ \(E\) admits a natural Ap -module struct (Budyak 125) - Furthermore, if E is a ring spectrum of ﬁnite Zor Z[p]-type then H ∗ \(E\) is a coalgebra over the Hopf algebra Ap , (Budyak 126) - d this homomorphism turns H∗ \(E\) into a comodule over the Hopf algebra A p∗ \. (Budyak 126) - 6\.29\. Theorem\. Let A be a connected Hopf algebra over a ﬁeld k, let C be a connected module coalgebra over A, and let v be the counit of C\. Let p : C → GC be the canonical epimorphism \(see 6\.5\), and let λ : GC → C be a k-homomorphism with pλ = 1GC \. Let Bm be the subspace of A ⊗ GC generated by all elements a⊗x with |a| ≤ m\. If the map ν : A → C, ν\(a\) = av, is monic for |a| ≤ m then the A-homomorphism η : A ⊗ GC → C, η\(a ⊗ x\) = a\(λx\) is monic on Bm \. Furthermore, η is epic and thus is an isomorphism in dimensions ≤ m\. Proof\. This theorem is a version of the Milnor–Moore Theorem, (Budyak 126) - 6\.30\. Corollary \(The Milnor–Moore Theorem\)\. Let A and C be as in 6\.29\. If ν : A → C, ν\(a\) = av, is monic then there is an isomorphism of A-modules C∼ = A ⊗ GC\. In particular, C is a free A-module (Budyak 127) - a canonical antiautomorphism c : A → A \(called also an antipode\) as follows (Budyak 128) - 6\.35\. Deﬁnition\. Given a connected Hopf algebra \(A, μ, η, Δ, ε\), we deﬁne a canonical antiautomorphism c : A → (Budyak 128) - Milnor–Moore [1]\. The ﬁrst example of such an antiautomorphism was found by Thom [1]; this was the canonical antiautomorphism χ : Ap → Ap of the Steenrod algebra Ap \. (Budyak 128) - Hopf [1] found that ordinary \(co\)homology rings of Lie groups \(in fact, H-spaces\) had certain speciﬁc algebraic properties\. \(For example, the rational cohomology ring of a Lie group is a free commutative algebra\.\) Afterwards Borel [1] clariﬁed the situation: every algebra A \(over a ﬁeld\) admitting a diagonal Δ : A → A ⊗ A, Δ\(ab\) = Δ\(a\)Δ\(b\), has such properties\. Borel suggested the name “Hopf algebra” for such object; (Budyak 129) - Milnor [2] discovered that the Steenrod algebra is a Hopf algebra\. In this way he got a new description of Ap , and this enabled him \(and some others\) to compute initial terms of certain Adams spectral sequences\. (Budyak 129) - Every graded abelian group G can be realized as the total homotopy group π∗ \(H\(G\)\) of the graded Eilenberg–Mac Lane spectrum H\(G\), but not every spectrum E is \(equivalent to\) the graded Eilenberg–Mac Lane spectrum H\(π∗ \(E\)\)\. (Budyak 130) - \. \(For example, the sphere spectrum S is not, because otherwise H∗ \(HZ\) would be a direct summand of H∗ \(S\)\.\) It is clear that it is useful to know whether a spectrum is a graded Eilenberg–Mac Lane spectrum\. (Budyak 130) - For example, Thom [2] proved that the spectrum M O of the non-oriented \(co\)bordism is a graded Eilenberg–Mac Lane spectrum, and this enabled him to compute the group π∗ \(M O\) \(i\.e\., non-oriented cobordism group\) and to prove the realizability of all Z/2-homology classes by singular manifold (Budyak 130) - In this section we give some suﬃcient conditions for a spectrum E to be a graded Eilenberg–Mac Lane spectrum, i\.e\., E H\(π∗ \(E\) (Budyak 130) - 7\.6\. Corollary\. Every HZ-module spectrum is a graded Eilenberg–Mac Lane spectrum\. (Budyak 132) - 7\.9\. Corollary\. For every spectrum E, the spectrum HZ ∧ E is a graded Eilenberg–Mac Lane spectrum\. (Budyak 132) - 7\.8\. Corollary\. If a ring spectrum E admits a ring morphism HZ → E then E is a graded Eilenberg–Mac Lane spectrum\. (Budyak 132) - 7\.11\. Theorem\. \(i\) The Q -localization of the sphere spectrum S is HQ\. In particular, the Hurewicz homomorphism h : π∗ \(E\) ⊗ Q → H∗ \(E\) ⊗ Q is an isomorphism for every spectrum E\. (Budyak 132) - 7\.13\. Theorem-Deﬁnition \(cf\. Dold [1]\)\. For every ring spectrum E there exists a ring equivalence E[0] → H\(π∗ \(E\) ⊗ Q\)\. This equivalence is called the Chern–Dold character with respect to E and is denoted by chE \. (Budyak 134) - 7\.14\. Theorem\. Let p be an odd prime, and let E be a Z[p]-local spectrum of ﬁnite Z[p]-type\. If E ∧ M \(Z/p\) is a graded Eilenberg–Mac Lane spectrum then so is E\. (Budyak 134) - 7\.16\. Theorem\. Let E be a spectrum of ﬁnite Z[p]-type\. If H ∗ \(E\) is a free Ap -module then E is a graded Eilenberg–Mac Lane spectrum\. (Budyak 136) - 7\.28\. Corollary\. If a commutative ring spectrum E is a graded Eilenberg– Mac Lane spectrum with pπ∗ \(E\) = 0, then there is an isomorphism H∗ \(E\) ∼ = A p∗ ⊗ π∗ \(E\) of Ap -comodule algebras\. (Budyak 141) - 7\.30\. Theorem \(Boardman [1]\)\. Let E, F be two commutative ring spectra\. Suppose that E, F are graded Eilenberg–Mac Lane spectra with pπ∗ \(E\) = 0 = pπ∗ \(F \)\. Then every ring homomorphism r : π∗ \(E\) → π∗ \(F \) is induced by a ring morphism f : E → F \. So, if there exists a ring isomorphism π ∗ \(E\) ∼ = ∼ = π∗ \(F \) then there exists a ring equivalence E F \. In particular, there is a ring equivalence E H\(π∗ \(E\)\)\. (Budyak 142) - A phantom, or a phantom map, is an essential map f : X → Y of a CW complex X such that f |X \(n\) is inessential for every n (Budyak 143) - 1\.1\. Deﬁnition\. A map f : X → Y of spaces \(or a morphism of spectra\) is called an X -phantom if σ[f ] = ∗ while [f ] = ∗\. Similarly, an element a ∈ E ∗ \(X\) is an X -phantom if σ\(a\) = 0 while a = 0\. This deﬁnition is given for an arbitrary family X , but really interesting are families with ∪Xα = X (Budyak 143) - The classes of X homotopic maps \(or X -equivalent elements\) form a set [X, Y ]X \(or a group E ∗ X \(X\)\) with the distinguished element ∗ given by a constant map \. There are the obvious quotient functions σ : [X, Y ] → [X, Y ]X and σ : E ∗ \(X\) → (Budyak 143) - E ∗ X \(X\)\. (Budyak 143) - 1\.5\. Example of a weak phantom\. Let X = S n [1/3] be a Z[1/3]-localized sphere S n , n > 1, i\.e\., the telescope of a sequence f f f f Sn − → Sn − → ··· − → Sn − → ··· , where f : S n → S n is a map of degree 3\. If we regard S n as a CW -complex with two cells, we obtain a cellular decomposition of X with 0-, nand \(n+1\)dimensional cells\. This gives us a chain complex {C∗ \(X\), ∂∗ }, where Cn \(X\) has Z-basis {a1 , \. \. \. , ai , \. \. \. }, and Cn+1 \(X\) has Z-basis {b1 , \. \. \. , bn , \. \. \. }, and ∂ n+1 bi = ai − 3ai+1 \. Let X = {Xλ } be the family of all ﬁnite CW n+1 subcomplexes of X\. It is clear that HX \(X\) = H n+1 \(S n \) = 0\. On the other hand, H n+1 \(X\) = 0 because the cocycle ϕ : Cn+1 \(X\) → Z, ϕ\(bi \) = 1 for every i, is not a coboundary\. Indeed, if ϕ = δψ for some ψ : Cn \(X\) → Z, then ψ\(ai \) − 3ψ\(ai+1 \) = 1\. In particular, (Budyak 144) - ψ\(a 1 \) = + 3k ψ\(ak+1 \) 2 for every k\. Hence, 3k divides 2ψ\(a1 \) + 1 for every k, and so 2ψ\(a1 \) = −1\. This is a contradiction\. Thus, the subgroup of weak phantoms of H n+1 \(X\) is nontrivial \(and even uncountable, see 5\.1 below\)\. (Budyak 145) - 1\.6\. Example of a phantom \(Adams–Walker [1]\)\. Let X = S 1 ∧ CP ∞ \. Consider the space T = S 3 [0], the telescope of the sequence S 13 ϕ1 ϕ2 ϕn −→ S23 −→ · · · − → Sn3 −−→ Sn+1 3 − → ··· , where Sn3 is a copy of S 3 and deg ϕn = n\. As in 1\.5, we have C3 \(T \) = {a1 , \. \. \. , an , \. \. \. }, C4 \(T \) = {b1 , \. \. \. , bn , \. \. \. } (Budyak 145)