Chapter 3

Use the universal coefficient theorem to show that if \(H_{*}(X ; \mathbb{Z})\) is finitely generated, so the Euler characteristic \(x(X)=\sum_{n}(-1)^{n} \operatorname{rank} H_{n}(X ; \mathbb{Z})\) is defined, then for any coefficient field \(F\) we have \(\chi(X)=\sum_{n}(-1)^{n} \operatorname{dim} H_{n}(X ; F)\)

Given maps \(f_{i}: X_{i} \rightarrow X_{i+1}\) for integers \(i<0,\) show that the 'reverse mapping telescope' obtained by glueing together the mapping cylinders of the \(f_{i}\) 's in the obvious way deformation retracts onto \(X_{0} .\) Similarly, if maps \(f_{i}: X_{i} \rightarrow X_{i+1}\) are given for all \(i \in \mathbb{Z},\) show that the resulting 'double mapping telescope' deformation retracts onto any of the ordinary mapping telescopes contained in it, the union of the mapping cylinders of the \(f_{i}\) 's for \(i\) greater than a given number \(n\).

Assuming as known the cup product structure on the torus \(S^{1} \times S^{1},\) compute the cup product structure in \(H^{*}\left(M_{g}\right)\) for \(M_{g}\) the closed orientable surface of genus \(g\) by using the quotient map from \(M_{g}\) to a wedge sum of \(g\) tori, shown below.

Show that there exist nonorientable 1 -dimensional manifolds if the Hausdorff condition is dropped from the definition of a manifold.

Show that \(\operatorname{Ext}(H, G)\) is a contravariant functor of \(H\) for fixed \(G,\) and a covariant functor of \(G\) for fixed \(H.\)

Show that \(\operatorname{Tor}(A, \mathbb{Q} / \mathbb{Z})\) is isomorphic to the torsion subgroup of \(A\). Deduce that \(A\) is torsionfree iff \(\operatorname{Tor}(A, B)=0\) for all \(B\)

Show that a retract of an H-space is an H-space if it contains the identity element.

Let \(C\) and \(C^{\prime}\) be chain complexes, and let \(I\) be the chain complex consisting of \(\mathbb{Z}\) in dimension 1 and \(\mathbb{Z} \times \mathbb{Z}\) in dimension \(0,\) with the boundary map taking a generator \(e\) in dimension 1 to the difference \(v_{1}-v_{0}\) of generators \(v_{i}\) of the two \(\mathbb{Z}\) 's in dimension 0. Show that a chain map \(f: I \otimes C \rightarrow C^{\prime}\) is precisely the same as a chain homotopy between the two chain maps \(f_{i}: C \rightarrow C^{\prime}, c \mapsto f\left(v_{i} \otimes c\right), i=0,1 .\) [The chain homotopy is \(h(c)=f(e \otimes c) .1\)

Using the cup product \(H^{k}(X, A ; R) \times H^{\ell}(X, B ; R) \rightarrow H^{k+\ell}(X, A \cup B ; R)\), show that if \(X\) is the union of contractible open subsets \(A\) and \(B\), then all cup products of positive-dimensional classes in \(H^{*}(X ; R)\) are zero. This applies in particular if \(X\) is a suspension. Generalize to the situation that \(X\) is the union of \(n\) contractible open subsets, to show that all \(n\) -fold cup products of positive-dimensional classes are zero.

Show that deleting a point from a manifold of dimension greater than 1 does not affect orientability of the manifold.

Let \(\mathcal{B}(X ; G)\) be the set of isomorphism classes of bundles of groups \(E \rightarrow X\) with fiber \(G,\) and let \(E_{0} \rightarrow B\) Aut \((G)\) be the bundle corresponding to the 'identity' action \(\rho: \operatorname{Aut}(G) \rightarrow \operatorname{Aut}(G) .\) Show that the map \([X, B \text { Aut }(G)] \rightarrow \mathcal{B}(X, G),[f] \mapsto f^{*}\left(E_{0}\right),\) is a bijection if \(X\) is a CW complex, where \([X, Y]\) denotes the set of homotopy classes of maps \(X \rightarrow Y\)

Show that if \(X\) is an H-space such that the set of path-components of \(X\) is a group with respect to the multiplication induced by the H-space structure, then all the path-components are homotopy equivalent.

Show that every covering space of an orientable manifold is an orientable manifold.

Regarding \(\mathbb{Z}_{2}\) as a module over the ring \(\mathbb{Z}_{4},\) construct a resolution of \(\mathbb{Z}_{2}\) by free modules over \(\mathbb{Z}_{4}\) and use this to show that \(\operatorname{Ext}_{z_{4}}^{n}\left(\mathbb{Z}_{2}, \mathbb{Z}_{2}\right)\) is nonzero for all \(n.\)

Using the cup product structure in \(H^{*}(S O(5) ; \mathbb{Z}),\) show that \(S O(5)\) is not homotopy equivalent to the product of any two CW complexes with nontrivial cohomology.

Show that if finite connected CW complexes \(X\) and \(Y\) are homotopy equivalent, then their universal covers \(\tilde{X}\) and \(\tilde{Y}\) are proper homotopy equivalent.

An abelian group \(G\) is defined to be divisible if the map \(G \stackrel{n}{\longrightarrow} G, g \mapsto n g,\) is surjective for all \(n>1 .\) Show that a group is divisible iff it is a quotient of a direct sum of \(\mathbb{Q}\) 's. Deduce from the previous problem that if \(G\) is divisible then \(\operatorname{Ext}(A, G)=0\) for all \(A\).

Given a covering space action of a group \(G\) on an orientable manifold \(M\) by orientation-preserving homeomorphisms, show that \(M / G\) is also orientable.

What happens if one defines homology groups \(h_{n}(X ; G)\) as the homology groups of the chain complex \(\cdots \rightarrow \operatorname{Hom}\left(G, C_{n}(X)\right) \rightarrow \operatorname{Hom}\left(G, C_{n-1}(X)\right) \rightarrow \cdots ?\) More specifically, what are the groups \(h_{n}(X ; G)\) when \(G=\mathbb{Z}, \mathbb{Z}_{m},\) and \(\mathbb{Q} ?\)

Show that if \((X, e)\) is an H-space then \(\pi_{1}(X, e)\) is abelian. [Compare the usual composition \(f \cdot g\) of loops with the product \(\mu(f(t), g(t))\) coming from the H-space multiplication \(\mu .]\)

Show that \(M \times N\) is orientable iff \(M\) and \(N\) are both orientable.

$$\text { Show that } \operatorname{Ext}\left(\mathbb{Z}_{p^{\infty}}, \mathbb{Z}_{p}\right) \approx \mathbb{Z}_{p}$$

Given two disjoint connected \(n\) -manifolds \(M_{1}\) and \(M_{2}\), a connected \(n\) -manifold \(M_{1} \geqslant M_{2},\) their connected sum, can be constructed by deleting the interiors of closed \(n\) -balls \(B_{1} \subset M_{1}\) and \(B_{2} \subset M_{2}\) and identifying the resulting boundary spheres \(\partial B_{1}\) and \(\partial B_{2}\) via some homeomorphism between them. (Assume that each \(B_{i}\) embeds nicely in a larger ball in \(M_{i} .\) ) (a) Show that if \(M_{1}\) and \(M_{2}\) are closed then there are isomorphisms \(H_{i}\left(M_{1} \neq M_{2} ; \mathbb{Z}\right) \approx\) \(H_{i}\left(M_{1} ; \mathbb{Z}\right) \oplus H_{i}\left(M_{2} ; \mathbb{Z}\right)\) for \(0

Show that homology groups \(H_{n}^{\ell f}(X ; G)\) can be defined using locally finite chains, which are formal sums \(\sum_{\sigma} g_{\sigma} \sigma\) of singular simplices \(\sigma: \Delta^{n} \rightarrow X\) with coefficients \(g_{\sigma} \in G,\) such that each \(x \in X\) has a neighborhood meeting the images of only finitely many \(\sigma\) 's with \(g_{\sigma} \neq 0 .\) Develop this homology theory far enough to show that for a locally compact \(\mathrm{CW}\) complex \(X, H_{n}^{\ell f}(X ; G)\) can be computed using infinite cellular chains \(\sum_{\alpha} g_{\alpha} e_{\alpha}^{n}\)

For a map \(f: M \rightarrow N\) between connected closed orientable \(n\) -manifolds with fundamental classes \([M]\) and \([N],\) the degree of \(f\) is defined to be the integer \(d\) such that \(f_{*}([M])=d[N],\) so the sign of the degree depends on the choice of fundamental classes. Show that for any connected closed orientable \(n\) -manifold \(M\) there is a degree \(1 \operatorname{map} M \rightarrow S^{n}.\)

Show that the functors \(h^{n}(X)=\operatorname{Hom}\left(H_{n}(X), \mathbb{Z}\right)\) do not define a cohomology theory on the category of CW complexes.

Show that for a Moore space \(M(G, n)\) the Bockstein long exact sequence in cohomology associated to the short exact sequence of coefficient groups \(0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0\) reduces to an exact sequence \(0 \rightarrow \operatorname{Hom}(G, A) \rightarrow \operatorname{Hom}(G, B) \rightarrow \operatorname{Hom}(G, C) \rightarrow \operatorname{Ext}(G, A) \rightarrow \operatorname{Ext}(G, B) \rightarrow \operatorname{Ext}(G, C) \rightarrow 0\)

For a map \(f: M \rightarrow N\) between connected closed orientable \(n\) -manifolds, suppose there is a ball \(B \subset N\) such that \(f^{-1}(B)\) is the disjoint union of of balls \(B_{i}\) each mapped homeomorphically by \(f\) onto \(B\). Show the degree of \(f\) is \(\Sigma_{i} \varepsilon_{i}\) where \(\varepsilon_{i}\) is +1 or -1 according to whether \(f: B_{i} \rightarrow B\) preserves or reverses local orientations induced from given fundamental classes \([M]\) and \([N].\)

Many basic homology arguments work just as well for cohomology even though maps go in the opposite direction. Verify this in the following cases: (a) Compute \(H^{i}\left(S^{n} ; G\right)\) by induction on \(n\) in two ways: using the long exact sequence of a pair, and using the Mayer-Vietoris sequence. (b) Show that if \(A\) is a closed subspace of \(X\) that is a deformation retract of some neighborhood, then the quotient map \(X \rightarrow X / A\) induces isomorphisms \(H^{n}(X, A ; G) \approx\) \(\widetilde{H}^{n}(X / A ; G)\) for all \(n\) (c) Show that if \(A\) is a retract of \(X\) then \(H^{n}(X ; G) \approx H^{n}(A ; G) \oplus H^{n}(X, A ; G)\)

Show that a \(p\) -sheeted covering space projection \(M \rightarrow N\) has degree \(\pm p,\) when \(M\) and \(N\) are connected closed orientable manifolds.

Show that if \(f: S^{n} \rightarrow S^{n}\) has degree \(d\) then \(f^{*}: H^{n}\left(S^{n} ; G\right) \rightarrow H^{n}\left(S^{n} ; G\right)\) is multiplication by \(d .\)

If \(T^{n}\) is the \(n\) -dimensional torus, the product of \(n\) circles, show that the Pontryagin ring \(H_{*}\left(T^{n} ; \mathbb{Z}\right)\) is the exterior algebra \(\Lambda_{Z}\left[x_{1}, \cdots, x_{n}\right]\) with \(\left|x_{i}\right|=1\).

Using cup products, show that every map \(s^{k+\ell} \rightarrow S^{k} \times S^{\ell}\) induces the trivial homomorphism \(H_{k+\ell}\left(S^{k+\ell}\right) \rightarrow H_{k+\ell}\left(S^{k} \times S^{\ell}\right),\) assuming \(k>0\) and \(\ell>0.\)

Show that the spaces \(\left(S^{1} \times \mathrm{CP}^{\infty}\right) /\left(S^{1} \times\left\\{x_{0}\right\\}\right)\) and \(S^{3} \times \mathrm{CP}^{\infty}\) have isomorphic cohomology rings with \(\mathbb{Z}\) or any other coefficients. IAn exercise for \(\$ 4 . \mathrm{L}\) is to show these two spaces are not homotopy equivalent.]

Describe \(H^{*}\left(\mathrm{CP}^{\infty} / \mathrm{CP}^{1} ; \mathbb{Z}\right)\) as a ring with finitely many multiplicative generators. How does this ring compare with \(H^{*}\left(S^{6} \times 00 \mathrm{P}^{\infty} ; \mathbb{Z}\right) ?\)

Show that the coproduct in the Hopf algebra \(H_{*}(X ; R)\) dual to \(H^{*}(X ; R)\) is induced by the diagonal map \(X \rightarrow X \times X, x \mapsto(x, x)\).

For an \(n\) -manifold \(M\) and a compact subspace \(A \subset M,\) show that \(H_{n}(M, M-A ; R)\) is isomorphic to the group \(\Gamma_{R}(A)\) of sections of the covering space \(M_{R} \rightarrow M\) over \(A\) that is, maps \(A \rightarrow M_{R}\) whose composition with \(M_{R} \rightarrow M\) is the identity.

Suppose that \(X\) is a path-connected H-space such that \(H^{*}(X ; \mathbb{Z})\) is free and finitely generated in each dimension, and \(H^{*}(X ; \mathbb{Q})\) is a polynomial ring \(\mathbb{Q}[\alpha]\). Show that the Pontryagin ring \(H_{*}(X ; \mathbb{Z})\) is commutative and associative, with a structure uniquely determined by the ring \(H^{*}(X ; \mathbb{Z})\).

Classify algebraically the Hopf algebras \(A\) over \(\mathbb{Z}\) such that \(A^{n}\) is free for each \(n\) and \(A \otimes \mathbb{Q} \approx \mathbb{Q}[\alpha] .\) In particular, determine which Hopf algebras \(A \otimes \mathbb{Z}_{p}\) arise from such \(A\) 's.

For the closed orientable surface \(M\) of genus \(g \geq 1,\) show that for each nonzero \(\alpha \in H^{1}(M ; \mathbb{Z})\) there exists \(\beta \in H^{1}(M ; \mathbb{Z})\) with \(\alpha \beta \neq 0 .\) Deduce that \(M\) is not homotopy equivalent to a wedge sum \(X \vee Y\) of \(C W\) complexes with nontrivial reduced homology. Do the same for closed nonorientable surfaces using cohomology with \(\mathbb{Z}_{2}\) coefficients.

Show that for a locally compact \(\Delta\) -complex \(X\) the simplicial and singular cohomology groups \(H_{c}^{i}(X ; G)\) are isomorphic. This can be done by showing that \(\Delta_{c}^{i}(X ; G)\) is the union of its subgroups \(\Delta^{i}(X, A ; G)\) as \(A\) ranges over subcomplexes of \(X\) that contain all but finitely many simplices, and likewise \(C_{c}^{i}(X ; G)\) is the union of its subgroups \(C^{i}(X, A ; G)\) for the same family of subcomplexes \(A.\)

Show that if a closed orientable manifold \(M\) of dimension \(2 k\) has \(H_{k-1}(M ; Z)\) torsionfree, then \(H_{k}(M ; \mathbb{Z})\) is also torsionfree.

Show that after a suitable change of basis, a skew-symmetric nonsingular bilinear form over \(\mathbb{Z}\) can be represented by a matrix consisting of \(2 \times 2\) blocks \(\left(\begin{array}{l}0 \\ 1 \\ 1\end{array}\right)\) along the diagonal and zeros elsewhere. [For the matrix of a bilinear form, the following operation can be realized by a change of basis: Add an integer multiple of the \(i^{t h}\) row to the \(j^{t h}\) row and add the same integer multiple of the \(i^{t h}\) column to the \(j^{t h}\) column. Use this to fix up each column in turn. Note that a skew-symmetric matrix must have zeros on the diagonal.]

28\. Show that a nonsingular symmetric or skew-symmetric bilinear pairing over a field \(F,\) of the form \(F^{n} \times F^{n} \rightarrow F,\) cannot be identically zero when restricted to all pairs of vectors \(v, w\) in a \(k\) -dimensional subspace \(V \subset F^{n}\) if \(k>n / 2.\)

Show that if \(M\) is a compact \(R\) -orientable \(n\) -manifold, then the boundary map \(H_{n}(M, \partial M ; R) \rightarrow H_{n-1}(\partial M ; R)\) sends a fundamental class for \((M, \partial M)\) to a fundamental class for \(\partial M.\)

Show that a compact manifold does not retract onto its boundary.

Show that if \(M\) is a compact contractible \(n\) -manifold then \(\partial M\) is a homology \((n-1)-\) sphere, that is, \(H_{i}(\partial M ; \mathbb{Z}) \approx H_{i}\left(S^{n-1} ; \mathbb{Z}\right)\) for all \(i.\)