6. 附录

4.3	(a) $\mathbb{Z}/m\oplus \mathbb{Z}/n$ contains an element $(n,m)$ of order $mn$, so $\mathbb{Z}/m\oplus \mathbb{Z}/n\simeq \mathbb{Z}/mn$. (b) $G\cong \mathbb{Z}/2\oplus \mathbb{Z}/9\oplus \mathbb{Z}/4\oplus \mathbb{Z}/9$. (c) $3,6=2\cdot 3,36=4\cdot 9$. (d) $30=3\cdot 2\cdot 5,180=9\cdot 4\cdot 5$.
6.4	(a) $H_1(X_n)={\mathbb{Z}}^{2n}$, $H_2(X_n)=\mathbb{Z}$. (b) $H_1(Y_n)={\mathbb{Z}}^{n-1}\oplus \mathbb{Z}/2$, $H_2(Y_n)=0$.
9.6	Using Exercise 39.2, $H_i(K,K_0)\simeq {\tilde{H}}_i(K/K_0)=\tilde{H}(\sigma /\mathrm{Bd}\sigma )\simeq {\tilde{H}}_i(S^2)=\{\begin{array}{ll}\mathbb{Z},& n=2\\ 0,& \text{else}\end{array}$
10.3	$H_i(S;\mathbb{Z}/3)=\{\begin{array}{ll}\mathbb{Z}/3,& i=0\\ \mathbb{Z}/3,& i=1\\ 0,& i=2\end{array}$. $H_i(S;\mathbb{Z}/4)=\{\begin{array}{ll}\mathbb{Z}/4,& i=0\\ \mathbb{Z}/4\oplus \mathbb{Z}/2,& i=1\\ \mathbb{Z}/2,& i=2\end{array}$.
14.1	$f(a)=A^{\prime },N^{\prime }$, $f(b)=B^{\prime }$, $f(c)=C^{\prime }$, $f(d)=C^{\prime }$, $f(e)=E^{\prime }$, $f(f)=E^{\prime },F^{\prime }$, $f(g)=F^{\prime },G^{\prime }$, $f(h)=G^{\prime },H^{\prime }$, $f(i)=I^{\prime }$, $f(j)=J^{\prime }$, $f(k)=K^{\prime },L^{\prime }$, $f(l)=L^{\prime }$, $f(m)=L^{\prime },M^{\prime }$, $f(n)=N^{\prime }$, $f(o)=O^{\prime }$, altogether $2^6=64$ simplicial approximations.
21.2	(a) Note that the ball $|z|\le c\simeq$ an interior point $0$, and so is image $h(|z|\le c)\simeq h(0)$ in ${\mathbb{R}}^2-0$. In particualr, the boundary $h(S_c)$ collapses into a point $h(0)$. It follows that $h_{\ast }\{S_c\}=h_{\ast }\{0\}=0$. (b) Set $F(z,t)=z^n+t(a_{n-1}{z}^{n-1}+\cdots +a_0)$. The image of $F$ never meets $0$ if $c$ is sufficiently large since $|F(z,t)|\ge c^n-t(|a_{n-1}|c^{n-1}+\cdots +|a_0|)>0$. This is a homotopy between $F(-,1)=h$ and $F(-,0)=k$. (c) $h_{\ast }=k_{\ast }:H_1(S_c)\to H_1({\mathbb{R}}^2-0)$ by (b), yet $h_{\ast }=0,k_{\ast }=n$.
22.4	(a) $F(-,t):M\to M$ for a fixed $t\in (-ϵ,ϵ)$ is a map with no fixed point, and $\simeq \mathrm{id}$ as $-$ varies from $t$ to $0$. Then $\chi (M)=\Lambda (\mathrm{id})=0$. (b) Exactly those with $\chi (M)=0$, i.e. the torus or the Klein bottle.
24.7	(a) For $|K|=P^2$ and $0\to \mathbb{Z}\to \mathbb{Z}\to \mathbb{Z}/2\to 0$, we haveFor $|K|=P^2$ and $0\to \mathbb{Z}/2\to \mathbb{Z}/4\to \mathbb{Z}/2\to 0$, we have (b) For $|K|=$ Klein and $0\to \mathbb{Z}\to \mathbb{Z}\to \mathbb{Z}/2\to 0$, we haveFor $|K|=$ Klein and $0\to \mathbb{Z}/2\to \mathbb{Z}/4\to \mathbb{Z}/2\to 0$, we have
25.3	$S^{n+1}$ is the suspension of $S^n$, and let $Sf:(w\ast S^n)/S^n\to (w\ast S^n)/S^n$ induced by $f$ be the suspension of $f$. The naturality of the boundary maps in the long exact sequence of the pair $(w\ast S^n,S^n)$ gives the commutative diagram, so $\mathrm{deg}Sf=\mathrm{deg}f$.The conclusion follows from induction and $k:S^1\to S^1,k(z)=z^n$ has degree $n$.
26.2	The three sequences of the same color are exact. From the exactness of $$H_p(A)\stackrel{i_{\ast }}{⟶}{H}_p(A)\stackrel{{\pi }_{\ast }}{⟶}{H}_p(A,A)\stackrel{{\partial }_{\ast }}{⟶}{H}_{p-1}(A)$$we deduce that $H_p(A,A)=0$, and due to the commutativity of (as all four maps are mere inclusions)we get $\eta \circ \pi =0$. Then apply the braid lemma in Exercise 1(b), we obtain the exactness of the sequence with black arrows in the above diagram.
26.3	[Hatcher] Section 2.2 Exercise 38, p.159.
29.1	(b) By (a), $H_p(\text{tsc})=\{\begin{array}{ll}{\mathbb{Z}}^2,& p=0\\ 0,& p>0\end{array}$.
31.3	$j:(X,A)\to (X,X)$ is a homotopy equivalence, so $H_p(X,A)=H_p(X,X)=0$.
31.5	[Hatcher] Theorem 2.10, p.111.$$\begin{array}{rl}\partial D_{{\Delta }_p}{i}_p& =\sum _{j\le i}(-1{)}^i(-1{)}^{j}l(({ϵ}_0,0),\dots ,(\widehat{{ϵ}_j,0}),\dots ,({ϵ}_i,0),({ϵ}_i,1),\dots ,({ϵ}_p,1))\\ & +\sum _{j\ge i}(-1{)}^i(-1{)}^{j+1}l(({ϵ}_0,0),\dots ,({ϵ}_i,0),({ϵ}_i,1),\dots ,(\widehat{{ϵ}_j,1}),\dots ,({ϵ}_p,1)).\end{array}$$The terms with $i=j$ in the two sums cancel except for $l((\widehat{{ϵ}_0,0}),({ϵ}_0,1),\dots ,({ϵ}_p,1))$, which is $j\circ i_p=j_{\#}(i_p)$, and $-l(({ϵ}_0,0),\dots ,({ϵ}_p,0),(\widehat{{ϵ}_p,1}))$, which is $-i\circ i_p=-i_{\#}(i_p)$. The terms with $i\ne j$ are exactly $-D_{{\Delta }_p}\partial i_p$ since$$\begin{array}{rl}{D}_{{\Delta }_p}\partial i_p& =\sum _{i<j}(-1{)}^i(-1{)}^{j}l(({ϵ}_0,0),\dots ,({ϵ}_i,0),({ϵ}_i,1),\dots ,(\widehat{{ϵ}_j,1}),\dots ,({ϵ}_p,1))\\ & +\sum _{j>i}(-1{)}^{i-1}(-1{)}^{j}l(({ϵ}_0,0),\dots ,(\widehat{{ϵ}_j,0}),\dots ,({ϵ}_i,0),({ϵ}_i,1),\dots ,({ϵ}_p,1)).\end{array}$$
33.1	Let $X_1$ be the arc minus the ends $(0,-1)$ and $(1,\sin 1)$, and $X_2$ be $X$ minus a point of $X_1$. $X_1,X_2$ is an open cover of $X$, yielding an exact sequence$$H_1(X_1)\oplus H_1(X_2)⟶H_1(X)\stackrel{\partial }{⟶}{H}_0(X_1\cap X_2)\stackrel{i}{⟶}{H}_0(X_1)\oplus H_0(X_2)$$All path components of $X_1$ and $X_2$ are contractible, so $H_1(X_1)\oplus H_1(X_2)=0$, and consequently $H_1(X)\simeq \mathrm{im}\partial =\ker i=0$. And $H_0(X)=\mathbb{Z}$ since $X$ is contractible.
33.3	$S^n$ is the sphere with equator $S^{n-1}$ and latitude $[-1,1]$, identifying $S^{n-1}\times 1$ to a north pole and $S^{n-1}\times (-1)$ to a south pole. Hence $H_p(S^n)=\{\begin{array}{ll}\mathbb{Z},& p=0,n\\ 0,& \text{else}\end{array}$.
35.4	(a) As $\overline{\mathrm{St}}v$ is contractible and all of its homology groups vanish except for $p=0$, for $p>0$ $H_p(\mathrm{Lk}v)\simeq H_p(\overline{\mathrm{St}}v-v)\simeq H_{p+1}(\overline{\mathrm{St}}v,\overline{\mathrm{St}}v-v)=\{\begin{array}{ll}\mathbb{Z},& p=n-1\\ 0,& \text{else}\end{array}$ when $x\in \mathrm{Int}M$ or $0$ when $x\in \mathrm{Bd}M$. (b) Suppose that there is a simplex $\sigma$ of dimension less than $m$ that is not a face of any simplex of dimension $m$. Then $\mathrm{Lk}\hat{\sigma }=\mathrm{Bd}\sigma$ is a sphere of dimension less than $m-1$, contradicting (a) for $v=\hat{\sigma }$ and $K\cup \mathrm{sd}\sigma$ in place of $K$. (c) $s$ cannot be a face of three simplices of maximal dimension, for otherwise the interiors of two of the simplices would overlap. If there are two $m$-simplices with $s$ being a face, then $\hat{s}$ is an interior point of $K$. Therefore $s$ is a face of precisely two $m$-simplices iff $\hat{s}$ is an interior point for any neighborhood of $\hat{s}$ in $K$, iff $h(s)\in \mathrm{Int}M$; on the contrary $s$ is a face of precisely one $m$-simplex iff $h(s)\in \mathrm{Bd}M$. so $s$ is a face of precisely two $m$-simplices iff $h(s)\in \mathrm{Bd}M$.
35.5	$S^3\simeq \mathrm{Bd}(B^2\times B^2)=(B^2\times \mathrm{Bd}{B}^2)\cup (\mathrm{Bd}{B}^2\times B^2)\simeq T_1\cup T_2$, and $T_1\cap T_2\simeq \mathrm{Bd}{B}^2\times \mathrm{Bd}{B}^2=S^1\times S^1=$ a hollow torus $T$, so
38.1	[Rotman] Lemma 8.18, Theorem 8.19, pp.201-202. Lemma. For a cell $e\in X$, the closure $\bar{e}$ lies in a finite CW subcomplex. Proof. We proceed by induction on $p=\dim e$; the statement is obviously true when $p=0$. If $p>0$, then Theorem 38.2(a) gives $\bar{e}-e=g(\mathrm{Bd}B)\subset X^{p-1}$. By “closure-finiteness”, $\bar{e}$ meets only finitely many cells other than $e$, say, $e_1,\dots ,e_m$, and we have just seen that $\dim e_i\le p-1$ for all $i$. By induction, there is a finite CW subcomplex $X_i$ containing ${\bar{e}}_i$, for $i=1,\dots ,m$, and each $X_i$ is closed. But $\bar{e}\subset e\cup X_1\cup \cdots \cup X_m$, so that this union of finitely many cells is closed and hence is a finite CW subcomplex. $\square$ (a) For each cell $e$ with $A\cap e\ne \varnothing$, choose a point $a_e\in A\cap e$, and let $E$ be the set compromised of all such $a_e$. For each cell $e$, the lemma says that there is a finite CW subcomplex $X_e$ containing $\bar{e}$. Therefore $E\cap \bar{e}\subset E\cap X_e$ is a finite set and hence is closed in $\bar{e}$. Since $X$ has the weak topology, $E$ is closed in $X$; indeed the same argument shows that every subset of $E$ is closed in $X$, hence $E$ is discrete. But $E$ is also compact, being a closed subset of $A$. Thus $E$ is finite, so that $A$ meets only finitely many cells $e$, say, $e_1,\dots ,e_m$. (b) By the lemma again, there are finite CW subcomplexes $X_i$ with ${\bar{e}}_i\subset X_i$ for $i=1,\dots ,m$. It follows that $A$ lies in the finite CW subcomplex $\bigcup _{1\le i\le m}{X}_i$.
38.4	(a) This is a special case of the proposition, having nothing to do with CW complexes, that $X\times Y$ is compactly generated if $X$ is compactly generated Hausdorff and $Y$ is locally compact. For the proof, cf. [Hatcher] Theorem A.15, p.531. (b) Let $\{f_{\alpha }:B_{\alpha }\to X{\}}_{\alpha }$ be the family of the characteristic maps of $X$, $\{g_{\beta }:B_{\beta }\to Y{\}}_{\beta }$ be the family of the characteristic maps of $X$. Then $\{(f_{\alpha },g_{\alpha }):B_{\alpha }\times B_{\beta }\to X\times Y{\}}_{\alpha ,\beta }$ is a CW complex structure on $X\times Y$. Observe that every compact set $A$ of $X\times Y$ lies in the product of its projections onto $X$ and $Y$, and the two projections are compact and hence lie in finite subcomplexes of $X$ and $Y$ according to Exercise 38.1, so $A$ lies in the image of a finite union of products $B_{\alpha }\times B_{\beta }$. Hence the prodoct CW structure generates $X{\times }_{C}Y$.
38.5	[Lu&Wei] Theorem 1.7, p.80. Let $X$ be a regular cell complex. Let $T(X)$ denote the abstract simplicial complex with vertices corresponding to the cells of $X$ and $n$-simplices members of the set $\{(e_0,e_1,\dots ,e_n)\mid e_i\text{ is a cell, }{e}_0⊊e_1⊊\cdots ⊊e_n\}$. Let $|T(X)|$ be the geometric realization of this complex. We define a homeomorphism $\Psi :|T(X)|\to X$ such that if $e$ is a cell of $X$, ${\Psi }^{-1}(e)$ is a subcomplex of $|T(X)|$. The map $\Psi$ is construted by first noticing that $T(X)=\bigcup _{n}T(X^n)$ and inductively constructing ${\Psi }_n:|T(X^n)|\to X^n$ so that ${\Psi }^n$ defines a triangulation of each cell of dimension less than $n+1$ which contains the triangulated boundary of the cell as a subcomplex. First observe that if ${\Psi }^{n-1}$ is defined and $e^n$ is an $n$-cell, then $\mathrm{Bd}{e}^n$ is a subcomplex of $X^{n-1}$, and $({\Psi }^{n-1}{)}^{-1}(\mathrm{Bd}{e}^n)$ is a subcomplex of $|T(X^n)|$ which is homeomorphic to the $(n-1)$-sphere. The usual conical construction enables us to extend ${\Psi }^{n-1}$ over $|T(X^n)|$. Geometrically, we are perforimg a generalized barycentric subdivision of the complex $X$. The new vertices correspond to the “barycenters” of the cells of $X$. The new cells correspond to the cones over the new $(n-1)$-cells of $X$ with vertices at the “barycenters” of the $n$-cells.
39.2	[Hatcher] Proposition 2.22, p.124. Let $V$ be a neighborhood of $A$ in $X$ that deformation retracts onto $A$. We have a commutative diagramThe upper left horizontal map is an isomorphism since in the long exact sequence of the triple $(X,V,A)$ the groups $H_p(V,A)$ are zero for all $p$, because a deformation retraction of $V$ onto $A$ gives a homotopy equivalence of pairs $(V,A)\simeq (A,A)$, and $H_p(A,A)=0$. The deformation retraction of $V$ onto $A$ induces a deformation retraction of $V/A$ onto $A/A$, so the same argument shows that the lower left horizontal map is an isomorphism as well. The other two horizontal maps are isomorphisms directly from excision. The right-hand vertical map $q$ is an isomorphism since $q$ restricts to a homeomorphism on the complement of $A$. From the commutativity of the diagram it follows that the left-hand $q_{\ast }$ is an isomorphism.
39.3	$D_p(A)$ is a free abelian group with a basis $\{(f_{\alpha }{)}_{\ast }(\gamma ):f_{\alpha }$ is a characteristic map for a $p$-cell of $A\}$, and inclusion simply maps each $p$-cell of $A$ to itself in $X$. (a) According to Five lemma, the isomorphisms between the homology groups of CW complexes and simplicial homology groups induce isomorphisms between the relative ones.(b) Let $X_p=X^p\cup A$ for all intergers $p$. Repeat the proof of Theorem 39.4, replacing $H_i(X_p)$ by $H_i(X_p,A)$ and $H_i(X)$ by $H_i(X,A)$, we get $H_i(X_p,X_{p-1})\simeq H_i(X,A)$ (for a detailed proof, see [Jiang] Section 3.2, 3.3, pp.99-104). Plus, the isomorphism between $\mathcal{D}(X,A)\simeq \underset{\alpha }{⨁}{H}_p(B_{\alpha }^p,S_{\alpha }^{p-1})$ and $H_p(X_p,X_{p-1})$ is given by$$\underset{\alpha }{⨁}{H}_p(B^p,S^{p-1})\stackrel{\underset{\alpha }{⨁}(f_{\alpha }{)}_{\ast }}{⟶}{H}_p(X_p,X_{p-1}),H_p(X_p,X_{p-1})\stackrel{\underset{\alpha }{⨁}(q_{\alpha }{)}_{\ast }}{⟶}\underset{\alpha }{⨁}{H}_p(B^p,S^{p-1}),$$where the index $\alpha$ is taken over all $p$-cells in $X-A$, $f_{\alpha }$ is the characteristic map for $e_{\alpha }^p$, and $q_{\alpha }$ is the quotient map collapsing $X^p-e_{\alpha }^p$ to a point.
39.4	(a) Let $\gamma$ be the map induced by the following diagramwhere $E_{-}^n$ is the hemisphere with the south pole $x_0$, $g$ is a homeomorphism, $h$ is a homotopy equivalance stretching $E_{+}^n$ to cover $S^n$, and the inclusion induces an isomorphism in homology since$$H_n(E_{+}^n,S^{n-1})\simeq H_n(S^n-U,E_{-}^n-U)\simeq H_n(S^n,E_{-}^n)$$where $U$ is a small disk neighborhood of $x_0$, and the first isomorphism is by homotopy equivalence and the second by excision. Suppose that $\varphi$ is multiplication by $d$, then it is the homomorphism induced by $\gamma$ composite a map from $S^n$ to $S^n$ of degree $d$. (b) This is because the inclusions induce an isomorphism $\oplus (i_{\alpha }{)}_{\ast }:\underset{\alpha }{⨁}{H}_n(S_{\alpha }^n,p)\to H_n(X,p)$. (c) For $p>0$, $B_{p-1}$, a subgroup of the free group $C_{p-1}$, is also free and thus projective. It follows that there is a homomorphism $k_{-1}:B_{q-1}\to C_q$ such that $\partial \circ k_{p-1}={\mathrm{id}}_{B_{p-1}}$, and therefore $C_q=k_{p-1}{B}_{p-1}\oplus Z_p$.For $p=0$, $H_0(C)=C_0/\partial C_1=C_0/\partial U_1$, and $A:=k_{-1}{H}_0(C)$ is non-trivial.(d) By (c), it suffices to construct a CW complex $X$ whose cellular chain complex is isomorphic to $\partial :U_n\to Z_{n-1}\to \mathrm{pt}$ for each $n$ (and $\partial U_1$ in place of $Z_{-1}$ for $n=0$) where $\mathrm{pt}$ is a point in $A$, since $\partial =0$ on $Z_n$ and $\mathrm{im}\partial \subset \ker \partial$. Both $U_n$ and $Z_{n-1}$ are free groups, bases denoted $\{x_{\alpha }\}$ and $\{y_{\beta }\}$, and write $\partial x_{\alpha }=\sum _{\beta }{d}_{\beta \alpha }{y}_{\beta }$. Let $X^n=\underset{\alpha }{\bigvee }{S}_{\alpha }^n$. We construct $X$ from $X^n$ by attaching cells $e_{\beta }^{n+1}$ via maps $f_{\beta }:S^n\to X^n$ such that the projection onto the summand $S_{\alpha }^n$ has degree $d_{\beta \alpha }$. Plus, generate $A$ by other points if $|A|>1$.

参考文献

[Hatcher]	A. Hatcher. Algebraic Topology. Cambridge University Press, 2002.
[Jiang]	姜伯驹. 同调论. 北京大学数学教学系列丛书. 北京大学出版社, 2007.
[Lu&Wei]	Lundell and Weigngram. The Topology of CW Complexes. The University Series in Higher Mathematics. Van Nonstrand Reinhold Company, 1969.
[Rotman]	J. J. Rotman. An Introduction to Algebraic Topology. Graduate Texts in Mathematics 119. Springer, 1998.