Chapter 2

What familiar space is the quotient \(\Delta\) -complex of a 2 -simplex \(\left[v_{0}, v_{1}, v_{2}\right]\) obtained by identifying the edges \(\left[v_{0}, v_{1}\right]\) and \(\left[v_{1}, v_{2}\right],\) preserving the ordering of vertices?

Show that the \(\Delta\) -complex obtained from \(\Delta^{3}\) by performing the edge identifications \(\left[v_{0}, v_{1}\right] \sim\left[v_{1}, v_{3}\right]\) and \(\left[v_{0}, v_{2}\right] \sim\left[v_{2}, v_{3}\right]\) deformation retracts onto a Klein bottle. Find other pairs of identifications of edges that produce \(\Delta\) -complexes deformation retracting onto a torus, a 2-sphere, and \(\mathbb{R P}^{2}\).

Given a map \(f: S^{2 n} \rightarrow S^{2 n}\), show that there is some point \(x \in S^{2 n}\) with either \(f(x)=x\) or \(f(x)=-x .\) Deduce that every map \(\mathbb{R P}^{2 n} \rightarrow \mathbb{R} P^{2 n}\) has a fixed point. Construct maps \(\mathrm{RP}^{2 n-1} \rightarrow \mathrm{RP}^{2 n-1}\) without fixed points from linear transformations \(\mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}\) without eigenvectors.

Verify that the formula \(f\left(z_{1}, \cdots, z_{2 k}\right)=\left(\bar{z}_{2},-\bar{z}_{1}, \bar{z}_{4},-\bar{z}_{3}, \cdots, \bar{z}_{2 k},-\bar{z}_{2 k-1}\right)\) defines a map \(f: \mathbb{C}^{2 k} \rightarrow \mathbb{C}^{2 k}\) inducing a quotient map \(\mathrm{CP}^{2 k-1} \rightarrow \mathrm{CP}^{2 k-1}\) without fixed points.

Let \(f: S^{n} \rightarrow S^{n}\) be a map of degree zero. Show that there exist points \(x, y \in S^{n}\) with \(f(x)=x\) and \(f(y)=-y .\) Use this to show that if \(F\) is a continuous vector field defined on the unit ball \(D^{n}\) in \(\mathbb{R}^{n}\) such that \(F(x) \neq 0\) for all \(x,\) then there exists a point on \(\partial D^{n}\) where \(F\) points radially outward and another point on \(\partial D^{n}\) where \(F\) points radially inward.

If \(X\) is a finite simplicial complex and \(f: X \rightarrow X\) is a simplicial homeomorphism, show that the Lefschetz number \(\tau(f)\) equals the Euler characteristic of the set of fixed points of \(f .\) In particular, \(\tau(f)\) is the number of fixed points if the fixed points are isolated. IHint: Barycentrically subdivide \(X\) to make the fixed point set a subcomplex.]

Compute the simplicial homology groups of the triangular parachute obtained from \(\Delta^{2}\) by identifying its three vertices to a single point.

Construct a surjective map \(S^{n} \rightarrow S^{n}\) of degree zero, for each \(n \geq 1\).

Let \(S\) be an embedded \(k\) -sphere in \(S^{n}\) for which there exists a disk \(D^{n} \subset S^{n}\) intersecting \(S\) in the disk \(D^{k} \subset D^{n}\) defined by the first \(k\) coordinates of \(D^{n} .\) Let \(D^{n-k} \subset D^{n}\) be the disk defined by the last \(n-k\) coordinates, with boundary sphere \(S^{n-k-1} .\) Show that the inclusion \(S^{n-k-1} \hookrightarrow S^{n}-S\) induces an isomorphism on homology groups.

Let \(M\) be a closed orientable surface embedded in \(\mathbb{R}^{3}\) in such a way that reflection across a plane \(P\) defines a homeomorphism \(r: M \rightarrow M\) fixing \(M \cap P,\) a collection of circles. Is it possible to homotope \(r\) to have no fixed points?

Show that any two reflections of \(S^{n}\) across different \(n\) -dimensional hyperplanes are homotopic, in fact homotopic through reflections. [The linear algebra formula for a reflection in terms of inner products may be helpful.]

Modify the construction of the Alexander horned sphere to produce an embedding \(S^{2} \hookrightarrow \mathbb{R}^{3}\) for which neither component of \(\mathbb{R}^{3}-S^{2}\) is simply-connected.

Let \(X\) be homotopy equivalent to a finite simplicial complex and let \(Y\) be homotopy equivalent to a finite or countably infinite simplicial complex. Using the simplicial approximation theorem, show that there are at most countably many homotopy classes of maps \(X \rightarrow Y\).

A polynomial \(f(z)\) with complex coefficients, viewed as a map \(\mathrm{C} \rightarrow \mathrm{C}\), can always be extended to a continuous map of one- point compactifications \(\hat{f}: S^{2} \rightarrow S^{2}\). Show that the degree of \(\hat{f}\) equals the degree of \(f\) as a polynomial. Show also that the local degree of \(\hat{f}\) at a root of \(f\) is the multiplicity of the root.

Compute the homology groups of the following 2-complexes: (a) The quotient of \(S^{2}\) obtained by identifying north and south poles to a point. (b) \(S^{1} \times\left(S^{1} \vee S^{1}\right)\) (c) The space obtained from \(D^{2}\) by first deleting the interiors of two disjoint subdisks in the interior of \(D^{2}\) and then identifying all three resulting boundary circles together via homeomorphisms preserving clockwise oricntations of these circles. (d) The quotient space of \(S^{1} \times S^{1}\) obtained by identifying points in the circle \(S^{1} \times\left\\{x_{0}\right\\}\) that differ by \(2 \pi / m\) rotation and identifying points in the circle \(\left\\{x_{0}\right\\} \times S^{1}\) that differ by \(2 \pi / n\) rotation.

Use the transfer sequence for the covering \(S^{\infty} \rightarrow \mathbb{R} P^{\infty}\) to compute \(H_{n}\left(\mathbb{R} P^{\infty} ; \mathbb{Z}_{2}\right)\)

Let \(X\) be the quotient space of \(S^{2}\) under the identifications \(x \sim-x\) for \(x\) in the equator \(S^{1} .\) Compute the homology groups \(H_{i}(X) .\) Do the same for \(S^{3}\) with antipodal points of the equatorial \(S^{2} \subset S^{3}\) identified.

Show that if \(A\) is a retract of \(X\) then the map \(H_{n}(A) \rightarrow H_{n}(X)\) induced by the inclusion \(A \subset X\) is injective.

Determine whether there exists a short exact sequence \(0 \rightarrow \mathbb{Z}_{4} \rightarrow \mathbb{Z}_{8} \oplus \mathbb{Z}_{2} \rightarrow \mathbb{Z}_{4} \rightarrow 0\) More generally, determine which abelian groups \(A\) fit into a short exact sequence \(0 \rightarrow \mathbb{Z}_{p^{m}} \rightarrow A \rightarrow \mathbb{Z}_{p^{m}} \rightarrow 0\) with \(p\) prime. What about the case of short exact sequences \(0 \rightarrow \mathbb{Z} \rightarrow A \rightarrow \mathbb{Z}_{n} \rightarrow 0 ?\)

A map \(f: S^{n} \rightarrow S^{n}\) satisfying \(f(x)=f(-x)\) for all \(x\) is called an even map. Show that an even map \(S^{n} \rightarrow S^{n}\) must have even degree, and that the degree must in fact be zero when \(n\) is even. When \(n\) is odd, show there exist even maps of any given even degree. IHints: If \(f\) is even, it factors as a composition \(S^{n} \rightarrow \mathbb{R} P^{n} \rightarrow S^{n} .\) Using the calculation of \(H_{n}\left(\mathbb{R} P^{n}\right)\) in the text, show that the induced map \(H_{n}\left(S^{n}\right) \rightarrow H_{n}\left(\mathbb{R P}^{n}\right)\) sends a generator to twice a generator when \(n\) is odd. It may be helpful to show that the quotient map \(\mathbb{R P}^{n} \rightarrow \mathbb{R} \mathrm{P}^{n} / \mathbb{R} \mathrm{P}^{n-1}\) induces an isomorphism on \(H_{n}\) when \(n\) is odd.]

(a) Show that \(H_{0}(X, A)=0\) iff \(A\) meets each path-component of \(X .\) (b) Show that \(H_{1}(X, A)=0\) iff \(H_{1}(A) \rightarrow H_{1}(X)\) is surjective and each path-component of \(X\) contains at most one path-component of \(A\).

Show the isomorphism between cellular and singular homology is natural in the following sense: \(A\) map \(f: X \rightarrow Y\) that is cellular \(-\) satisfying \(f\left(X^{n}\right) \subset Y^{n}\) for all \(n-\) induces a chain map \(f_{*}\) between the cellular chain complexes of \(X\) and \(Y,\) and the map \(f_{*}: H_{n}^{C W}(X) \rightarrow H_{n}^{C W}(Y)\) induced by this chain map corresponds to \(f_{*}: H_{n}(X) \rightarrow H_{n}(Y)\) under the isomorphism \(H_{n}^{C W} \approx H_{n}\).

Show that for the subspace \(\mathbb{Q} \subset \mathbb{R},\) the relative homology group \(H_{1}(\mathbb{R}, \mathbb{Q})\) is free abelian and find a basis.

Compute the homology groups of the subspace of \(I \times I\) consisting of the four boundary edges plus all points in the interior whose first coordinate is rational.

Compute \(H_{i}\left(\mathrm{RP}^{n} / \mathrm{RP}^{m}\right)\) for \(m

For finite CW complexes \(X\) and \(Y,\) show that \(\chi(X \times Y)=\chi(X) \chi(Y)\).

Prove by induction on dimension the following facts about the homology of a finite-dimensional CW complex \(X,\) using the observation that \(X^{n} / X^{n-1}\) is a wedge sum of \(n\) -spheres: (a) If \(X\) has dimension \(n\) then \(H_{i}(X)=0\) for \(i>n\) and \(H_{n}(X)\) is free. (b) \(H_{n}(X)\) is free with basis in bijective correspondence with the \(n\) -cells if there are no cells of dimension \(n-1\) or \(n+1\) (c) If \(X\) has \(k\) n-cells, then \(H_{n}(X)\) is generated by at most \(k\) elements.

For \(X\) a finite \(\mathrm{CW}\) complex and \(p: \tilde{X} \rightarrow X\) an \(n\) -sheeted covering space, show that \(\chi(\tilde{X})=n \chi(X)\).

Show that the second barycentric subdivision of a \(\Delta\) -complex is a simplicial complex. Namely, show that the first barycentric subdivision produces a \(\Delta\) -complex with the property that each simplex has all its vertices distinct, then show that for a \Delta-complex with this property, barycentric subdivision produces a simplicial complex.

Show that for each \(n \in \mathbb{Z}\) there is a unique function \(\varphi\) assigning an integer to each finite CW complex, such that (a) \(\varphi(X)=\varphi(Y)\) if \(X\) and \(Y\) are homeomorphic, (b) \(\varphi(X)=\varphi(A)+\varphi(X / A)\) if \(A\) is a subcomplex of \(X,\) and \((c) \varphi\left(S^{0}\right)=n .\) For such a function \(\varphi,\) show that \(\varphi(X)=\varphi(Y)\) if \(X \simeq Y\).

The short exact sequences \(0 \rightarrow C_{n}(A) \rightarrow C_{n}(X) \rightarrow C_{n}(X, A) \rightarrow 0\) always split, but why does this not always yield splittings \(H_{n}(X) \approx H_{n}(A) \oplus H_{n}(X, A) ?\)

Show that \(S^{1} \times S^{1}\) and \(S^{1} \vee S^{1} \vee S^{2}\) have isomorphic homology groups in all dimensions, but their universal covering spaces do not.

Use the Mayer-Vietoris sequence to show there are isomorphisms \(\tilde{H}_{n}(X \vee Y) \approx\) \(\tilde{H}_{n}(X) \oplus \tilde{H}_{n}(Y)\) if the basepoints of \(X\) and \(Y\) that are identified in \(X \vee Y\) are defor mation retracts of neighborhoods \(U \subset X\) and \(V \subset Y\).

For \(S X\) the suspension of \(X,\) show by a Mayer-Vietoris sequence that there are isomorphisms \(\tilde{H}_{n}(S X) \approx \tilde{H}_{n-1}(X)\) for all \(n\).

Suppose the space \(X\) is the union of open sets \(A_{1}, \cdots, A_{n}\) such that each intersection \(A_{i_{1}} \cap \cdots \cap A_{i_{k}}\) is either empty or has trivial reduced homology groups. Show that \(\tilde{H}_{i}(X)=0\) for \(i \geq n-1,\) and give an example showing this inequality is best possible, for each \(n\).

Derive the long exact sequence of a pair \((X, A)\) from the Mayer-Vietoris sequence applied to \(X \cup C A,\) where \(C A\) is the cone on \(A\). IWe showed after the proof of Proposition 2.22 that \(H_{n}(X, A) \approx \tilde{H}_{n}(X \cup C A)\) for all \(n .1\).

Use the Mayer-Vietoris sequence to show that a nonorientable closed surface, or more generally a finite simplicial complex \(X\) for which \(H_{1}(X)\) contains torsion, cannot be embedded as a subspace of \(\mathrm{R}^{3}\) in such a way as to have a neighborhood homeomorphic to the mapping cylinder of some map from a closed orientable surface to \(X .\) IThis assumption on a neighborhood is in fact not needed if one deduces the result from Alexander duality in \(\$ 3.3 .\rfloor\).

36\. Show that \(H_{i}\left(X \times S^{n}\right) \approx H_{i}(X) \oplus H_{i-n}(X)\) for all \(i\) and \(n,\) where \(H_{i}=0\) for \(i<0\) by definition. Namely, show \(H_{i}\left(X \times S^{n}\right) \approx H_{i}(X) \oplus H_{i}\left(X \times S^{n}, X \times\left[x_{0}\right]\right)\) and \(H_{i}\left(X \times S^{n}, X \times\left\\{x_{0}\right\\}\right) \approx H_{i-1}\left(X \times S^{n-1}, X \times\left\\{x_{0}\right\\}\right) .\) IFor the latter isomorphism the relative Mayer-Vietoris sequence yields an easy proof.|

Give an elementary derivation for the Mayer-Vietoris sequence in simplicial homology for a \(\Delta\) -complex \(X\) decomposed as the union of subcomplexes \(A\) and \(B\).

From the long exact sequence of homology groups associated to the short exact sequence of chain complexes \(0 \rightarrow C_{i}(X) \stackrel{n}{\longrightarrow} C_{i}(X) \rightarrow C_{i}\left(X ; \mathbb{Z}_{n}\right) \rightarrow 0\) deduce immediately that there are short exact sequences $$ 0 \rightarrow H_{i}(X) / n H_{i}(X) \rightarrow H_{i}\left(X ; Z_{n}\right) \rightarrow n-\text {Torsion}\left(H_{i-1}(X)\right) \rightarrow 0 $$ where \(n\) -Torsion(G) is the kernel of the map \(G \stackrel{n}{\longrightarrow} G, g \mapsto n g .\) Use this to show that \(\tilde{H}_{i}\left(X ; \mathbb{Z}_{p}\right)=0\) for all \(i\) and all primes \(p\) iff \(\tilde{H}_{i}(X)\) is a vector space over \(\mathbb{Q}\) for all \(i\).

For \(X\) a finite \(\mathrm{CW}\) complex and \(F\) a field, show that the Fuler characteristic \(\chi(X)\) can also be computed by the formula \(x(X)=\Sigma_{n}(-1)^{n} \operatorname{dim} H_{n}(X ; F),\) the alternating sum of the dimensions of the vector spaces \(H_{n}(X ; F)\).

Let \(X\) be a finite connected graph having no vertex that is the endpoint of just one edge, and suppose that \(H_{1}(X ; \mathbb{Z})\) is free abelian of rank \(n>1,\) so the group of automorphisms of \(H_{1}(X ; \mathbb{Z})\) is \(G L_{n}(Z),\) the group of invertible \(n \times n\) matrices with integer entries whose inverse matrix also has integer entries. Show that if \(G\) is a finite group of homeomorphisms of \(X\), then the homomorphism \(G \rightarrow G L_{n}(Z)\) assigning to \(g: X \rightarrow X\) the induced homomorphism \(g_{*}: H_{1}(X ; Z) \rightarrow H_{1}(X ; Z)\) is injective. Show the same result holds if the coefficient group \(\mathbb{Z}\) is replaced by \(\mathbb{Z}_{m}\) with \(m>2 .\) What goes wrong when \(m=2 ?\)

(a) Show that a chain complex of free abelian groups \(C_{n}\) splits as a direct sum of subcomplexes \(0 \rightarrow L_{n+1} \rightarrow K_{n} \rightarrow 0\) with at most two nonzero terms. IShow the short exact sequence \(0 \rightarrow \operatorname{Ker} \partial \rightarrow C_{n} \rightarrow \operatorname{Im} \partial \rightarrow 0\) splits and take \(K_{n}=\) Ker \(\partial .\) (b) In case the groups \(C_{n}\) are finitely generated, show there is a further splitting into summands \(0 \rightarrow \mathbb{Z} \rightarrow 0\) and \(0 \rightarrow \mathbb{Z} \stackrel{m}{\longrightarrow} \mathbb{Z} \rightarrow 0\). [Reduce the matrix of the boundary map \(L_{n+1} \rightarrow K_{n}\) to echelon form by elementary row and column operations. (c) Deduce that if \(X\) is a CW complex with finitely many cells in each dimension, then \(H_{n}(X ; G)\) is the direct sum of the following groups: \- a copy of \(G\) for each \(\mathbb{Z}\) summand of \(H_{n}(X)\) \- a copy of \(G / m G\) for each \(\mathbb{Z}_{m}\) summand of \(H_{n}(X)\) \- a copy of the kemel of \(G \stackrel{m}{\longrightarrow} G\) for each \(\mathbb{Z}_{m}\) summand of \(H_{n-1}(X)\)