《代数拓扑学基础》第一章习题选做

Chapter 1 Homology Groups of a Simplicial Complex

\(\newcommand{\CC}{\mathcal C} \newcommand{\RR}{\mathbb R} \newcommand{\SS}{\mathcal S} \newcommand{\QQ}{\mathbb Q} \newcommand{\ZZ}{\mathbb Z} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Bd}{Bd} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\St}{St} \DeclareMathOperator{\cSt}{\overline{St}}\)

§1 Simplicies

5. Let \(U\) be a bounded open set in \(\RR^n\). Suppose \(U\) is star-convex relative to the origin; this means that for each \(x\) in \(U\), the line segment from \(0\) to \(x\) lies in \(U\).
(a) Show that a ray from \(0\) may intersect \(\Bd U\) in more than one point.

Proof. Let \(U = \Int B^n - (0,\dots,0) \times [1/2, 1]\), i.e., removing a segment from the open ball. Then we have such segment lies in \(\Bd U\), and the ray from \(0\) intersects the whole segment. \(\square\)

§2 Simplicial Complexes and Simplicial Maps

7. Let \(K\) be a complex. Show that \(|K|\) is metrizable if and only if \(K\) is locally finite.

Proof. If \(K\) is locally finite, consider the function

\[d(x, y) = \sup_v |t_v(x) - t_v(y)|, \]

restrict to any simplex \(\sigma\), then for \(x, y\in \sigma\), we have

\[d(x, y) = \max_{v\in \sigma} |t_v(x) - t_v(y)|, \]

which clearly gives a metric of the topology of \(\sigma\). Thus the metric gives the topology of \(|K|\). The triangle inequality \(d(x,y) \leq d(x,z)+d(y,z)\) is also easy to prove since for each \(x,y,z\), the involving \(v\) are finite.
Inversely, if \(K\) is not locally finite and suppose \(|K|\) is metrizable, then it has a vertex \(v\) adjacent to infinite number of simplicies, say, \(\{\sigma_1,\dots,\sigma_n,\dots\} \cup \{\sigma'_i\}_{i\in J}\). For each \(\sigma_i\), let \(C_i = \sigma_i \cap B(v, 1/i)\). For each \(\sigma_i'\), and take

\[U = \left[\left(\bigcup_{n\geq 1} C_n\right) \cup \left(\bigcup_{i\in J} \sigma'_i\right)\right] \cap \St v, \]

One can verify that \(U\) is open on each \(\sigma\), thus \(U\) is open. But we cannot choose any \(\epsilon>0\) such that \(B(v, \epsilon) \subset U\), contradiction. \(\square\)

§3 Abstract Simplicial Complexes

3. Prove Lemma 3.2.

Proof. For each \(\sigma\), suppose \(v, w \in \sigma\) are different vertices, since \(\sigma \in \cSt v \cap \cSt w\), then \(v, w\) must have different labels. Therefore \(\dim g(\sigma) = \dim \sigma\), this proves the first claim.
For the second claim, suppose \(g(\sigma_1) = g(\sigma_2)\), then this induces pairs \(g(v_i) = g(w_i)\), since there is at least one pair of \(v_i \neq w_i\), the disjointness of \(\cSt v_i\) and \(\cSt w_i\) implies \(v_i=w_i\) for all \(i\), therefore, they should all belongs to \(L_0\). \(\square\)

§4 Review of Abelian Groups

Omitted.

§5 Homology Groups

1. Let \(\SS\) be the abstract complex consisting of the \(1\)-simplices \(\{v_0,v_1\},\{v_1,v_2\}, \dots,\{v_{n-1},v_n\},\{v_n,v_0\}\) and their vertices. If \(K\) is a geometric realization of \(\SS\), compute \(H_1(K)\).

Let \(C_1(K) = \sum_{i=0}^n m_i e_i\). Since there is no \(2\)-simplex, we have \(B_1(K)\) is trivial. And we have \(\sum m_ie_i \in Z_1(K)\) iff \(m_0=m_1=\cdots=m_n\), thus \(H_1(K)\cong \ZZ\).

6. An "infinite \(p\)-chain" on \(K\) is a function \(c\) from the oriented \(p\)-simplices of \(K\) to the integers such that \(c(\sigma) = - c (\sigma')\) if \(\sigma\) and \(\sigma'\) are opposite orientations of the same simplex. We do not require that \(c(\sigma) = 0\) for all but finitely many oriented simplices. Let \(C^\infty(K)\) denote the group of infinite \(p\)-chains. It is abelian, but it will not in general be free.
(a) Show that if \(K\) is locally finite, then one can define a boundary operator

\[\partial_p^\infty\colon C_p^\infty(K) \to C_{p-1}^\infty(K) \]

by the formula used earlier, and Lemma 5.3 holds. The resulting groups

\[H_p^\infty(K) = \ker \partial_p^{\infty} / \im \partial_{p+1}^{\infty} \]

are called the homology groups based on infinite chains.

Proof. When \(K\) is locally finite, the summation is finite at each simplex, the argument is similar. \(\square\)

(b) Let \(K\) be the complex whose space is \(\RR\) and whose vertices are the integers. Show that

$H_1(K) = 0\quad$ and $\quad H_1^\infty(K)=\ZZ$.

Proof. Let \(c = \sum_n m_n [n,n+1]\), for \(Z_1(K)\) we have \(\cdots=m_{-1}=m_0=m_1=\cdots\), so there is no nontrivial solution in \(C_1(K)\), but has solution \(m_n = k, k\in \ZZ\) for \(Z_1^\infty(K)\). The boundary are both trivial, thus we have the conclusion. \(\square\)

7. Let \(\SS\) be the abstract complex whose simplices consist of the sets \(\{im,m\}\), \(\{im,-m\}\), and \(\{m,-m\}\) for all positive integers \(m\), along with their faces. If \(K\) is a geometric realization of \(\SS\), compute \(H_1(K)\) and \(H_1^\infty(K)\).

For each fixed \(m\geq 1\) it gives an \(\ZZ\), so we have \(H_1(K)=\bigoplus_m \ZZ\), while \(H_1^\infty(K) = \prod_m \ZZ\).

§6 Homology Groups of Surfaces

2. The connected sum \(T \# T\) of two tori is obtained by deleting an open disc from each of two disjoint tori and gluing together the pieces that remain, along their boundaries. It can be represented as a quotient space of an octagonal region in the plane by making identifications on the boundary as indicated in Figure 6.10. (Splitting this octagon along the dotted line gives two tori with open discs deleted.)
(b) Compute the homology of \(T \# T\) in dimensions \(1\) and \(2\) by following the pattern of Theorem 6.2. Specifically, let \(A\) be the image of \(\Bd L\) under the quotient map; then \(A\) is a wedge of four circles. Orient each \(2\)-simplex of \(L\) counterclockwise; let \(\gamma\) be the sum of the correspondingly oriented simplices of \(K\). Show first that every \(1\)-cycle of \(K\) is homologous to one carried by \(A\). Then show that every \(2\)-chain of \(K\) whose boundary is carried by \(A\) is a multiple of \(\gamma\). Complete the computation by analyzing the \(1\)-cycles carried by \(A\), and by computing \(\partial \gamma\).

Similar to the argument of Theorem 6.2, we first have the two following:
(1) Every \(1\)-cycle of \(T\# T\) is homologous to a \(1\)-cycle carried by \(A\).
(2) If \(d\) is a \(2\)-chain of \(T\# T\) and if \(\partial d\) is carried by \(A\), then \(d\) is a multiple of \(\partial \gamma\).
Now assume the \(1\)-cycle is carried by \(A\), denoted by \(n_a A + n_bB + n_cC + n_d D\), since all the vertices of the boundary are glued together, we have no further restriction of the cycle. Thus \(H_1(T\# T) = \ZZ^4\).
Note that \(\partial \gamma = 0\), we have \(H_2(T\# T) = \ZZ\).

3. Compute the homology of \(4\)-fold connected sum \(P^2 \# P^2 \# P^2 \# P^2\) in dimensions \(1\) and \(2\).

We can still draw it on an octagon, with all edges counterclockwise, with label \(A,A,B,B,C,C,D,D\).
We still have all \(n_aA+n_bB +n_cC +n_dD\) is a cycle, but now we have \(\partial \gamma = 2A+2B+2C+2D\). Take a basis \(\{A,B,C,A+B+C+D\}\), we have \(H_1(P^2 \# P^2 \# P^2 \# P^2)=\ZZ^3\oplus \ZZ/2\), and \(H_2(P^2 \# P^2 \# P^2 \# P^2) = 0\).

7. Compute the homology of the space indicated in Figure 6.13.

Let the two faces be \(\gamma_A, \gamma_B\). A \(1\)-cycle should have form \(aA+bB+cC\), where \(C\) is the remain cycle. We have \(\partial \gamma_A = 3A, \partial \gamma_B = 3B\), we have \(H_1(K) = \ZZ \oplus (\ZZ/3)^2\), and \(H_2(K)=0\).

8. Given finitely generated abelian groups \(G_1\), and \(G_2\), with \(G_2\) free, show there is a finite \(2\)-dimensional complex \(K\) such that \(|K|\) is connected, \(H_1(K)\cong G_1\), and \(H_2(K)\cong G_2\).

Proof. A \(k\)-fold dunce cap provides a factor \(\ZZ/k\) in \(H_1\) and \(0\) in \(H_2\). A cycle without \(2\)-simplices provides a factor \(\ZZ\) in \(H_1\) and \(0\) in \(H_2\). A sphere \(S^2\) provides a factor \(0\) in \(H_1\) and \(\ZZ\) in \(H_2\). Glueing them at together at one point can give any finitely generated \(G_1, G_2\) as homology groups. \(\square\)

§7 Zero-Dimensional Homology

1. (a) Let \(G\) be an abelian group and let \(\phi\colon G\to \ZZ\) be an epimorphism. Show that \(G\) has an infinite cyclic subgroup \(H\) such that

\[G = (\ker \phi) \oplus H. \]

Proof. Since \(\phi\) is an epimorphism, we have some element \(\phi(x) = 1\), let \(H=\langle x\rangle\). For each \(g\in G\), we have \(n = \phi(g)\), thus \(g - nx \in \ker \phi\), the mapping \(g \mapsto (g - \phi(g)x, \phi(g)x)\) gives the isomorphism between \(G\) and \((\ker \phi) \oplus H\). \(\square\)

(b) Show that if \(\phi\colon C_0(K)\to \ZZ\) is any epimorphism such that \(\phi \circ \partial_1=0\), then

\[H_0(K) \cong (\ker \phi) / (\im \partial_1) \oplus \ZZ. \]

Proof. By the preceding argument, we have \(H_0(K) = C_0(K)/\im \partial_1 = (\ker \phi \oplus H)/\im \partial_1\), since \(\im \partial_1 \subset \ker \phi\), we have

\[H_0(K) = (\ker \phi \oplus H)/(\im \partial_1\oplus 0) = (\ker \phi / \im \partial_1) \oplus H. \square \]

§8 The Homology of a Cone

1. Let \(K\) be a complex; let \(w_0 *K\) and \(w_1 * K\) be two cones on \(K\) whose polytopes intersect only in \(|K|\).
(b) Using the bracket notation, define \(\phi \colon C_p(K)\to C_{p+1}(S(K))\) by the equation

\[\phi(c_p) = [w_0, c_p] - [w_1, c_p]. \]

Show that \(\phi\) induces a homomorphism

\[\phi_* \colon \tilde H_p(K) \to \tilde H_{p+1} (S(K)). \]

Proof. We first show that \(\phi\) maps \(\tilde Z_p(K)\) to \(\tilde Z_{p+1}(S(K))\). Suppose \(c_p\) is a cycle, then for \(p = 0\), we have \(\epsilon(c_p)=0\), thus

\[\partial \phi(c_p) = \partial[w_0, c_p] - \partial[w_1, c_p] = \epsilon(c_p)(w_0-w_1) = 0. \]

For \(p > 0\), we have \(\partial c_p=0\), thus

\[\partial \phi(c_p) = \partial[w_0, c_p] - \partial[w_1, c_p] = [w_0, \partial c_p] - [w_1, \partial c_p] = 0. \]

In conclusion, \(\phi(\tilde Z_p(K))\subset \tilde Z_{p+1}(S(K))\).
On another hand, for a boundary \(\partial c_{p+1}\), we have

\[ \phi(\partial c_{p+1}) = [w_0, \partial c_{p+1}] - [w_1, \partial c_{p+1}] = \partial([w_0, c_{p+1}] - [w_1, c_{p+1}]) \]

also bounds, thus \(\phi(B_p(K)) \subset B_{p+1}(S(K))\). This induces the homomorphism \(\phi_*\). \(\square\)

§9 Relative Homology

1. Let \(K\) be the complex pictured in Figure 9.2; let \(K_1\) be its ``outer edge". Compute \(H_i(K)\) and \(H_i(K,K_1)\).

Since \(K\) is connected, \(H_0(K) = \ZZ\). One can ''push'' the \(1\)-chains to the outer and inner edge, thus \(H_1(K)\) is generated by \(I-O\), the inner cycle \(I\) and the outer cycle \(O\), thus \(H_1(K) = \ZZ\). \(H_2(K)=0\).

Note that \(\im \partial\) in \(C_0(K)/C_0(K_1)\) is the whole group, we have \(H_0(K,K_1)=0\). \(H_1(K,K_1)\) is generated by \(I+C_1(K_1)\), thus \(H_1(K,K_1)=\ZZ\). \(H_2(K,K_1)=0\).

3. Show that if \(K\) is a complex and \(v\) is a vertex of \(K\), then \(H_i(K,v) \cong \tilde H_i(K)\) for all \(i\).

Proof. Since \(C_p(K)/C_p(v) = C_p(K)\) for \(p > 0\), this isomorphism automatically holds for \(i > 1\). The case \(i=0\) follows from \S7.1, because \(\sum n_w w \mapsto n_v\) is an epimorphism. For \(i=1\), if \(\partial c_1 \in C_0(v)\), we must have \(\partial c_1 = 0\), thus the \(\ker \partial_1\) is same for \(C_p(K,v)\) and \(C_p(K)\), thus the first homology is same. \(\square\)

4. Describe \(H_0(K,K_0)\) in general.

\(H_0(K, K_0)\) is a free abelian group, the rank is the number of connected components that are disjoint with \(K_0\).

§10 Homology with Arbitrary Coefficients

1. Compute the homology of \(P^2\) with \(\ZZ/2\) and \(\QQ\) coefficients.

Since \(P^2\) is connected, we have \(H_0(P^2;\ZZ/2)=\ZZ/2\), \(H_0(P^2; \QQ)= \QQ\).

The \(1\)-simplices \(a\) and \(b\) generates \(Z_1(P^2)\), but the boundaries are \(2a+2b\), thus \(H_1(P^2;\ZZ/2)=\ZZ/2\), and \(H_1(P^2;\QQ)=\QQ\). And \(H_2(P^2; \ZZ/2) = \ZZ/2\), and \(H_2(P^2;\QQ) = 0\).

§11 The Computability of Homology Groups

Omitted.

§12 Homomorphisms Induced by Simplicial Maps

5. Let \(f,g\colon (K,K_0)\to (L,L_0)\) be simplicial maps. Show that if \(f\) and \(g\) are contiguous as maps of \(K\) into \(L\), and if \(L_0\) is a full subcomplex of \(L\), then \(f\) and \(g\) are contiguous as maps of pairs.

Proof. For each simplex \(\sigma\in K_0\), we have the vertices of \(f(\sigma), g(\sigma)\) span a complex \(\tau\in L\). Since the vertices of \(f(\sigma), g(\sigma)\) all lies in \(L_0\) and \(L_0\) is a full subcomplex, we have \(\tau\in L_0\). Therefore, \(f,g\) restricting to \(K_0\to L_0\) are also contiguous, thus \(f,g\) are contiguous as maps of pairs. \(\square\)

§13 Chain Complexes and Acyclic Carriers

1. If \(\{\CC, \epsilon\}\) is an augmented chain complex, show that \(\tilde H_{-1}(\CC)=0\) and

\[\tilde H_0(\CC) \oplus \ZZ \cong H_0(\CC). \]

Conclude that \(\{\CC,\epsilon\}\) is acyclic if and only if \(H_p(\CC)\) is infinite cyclic for \(p = 0\) and vanishes for \(p\neq 0\).

Proof. Since \(\epsilon\) is an epimorphism, we have \(\ker \partial_{-1} = \ZZ = \im \epsilon\), thus \(\tilde H_{-1}(\CC) = 0\). Plug in \S7.1 (b) we have \(H_0(\CC)\cong (\ker \epsilon)/(\im \partial_1)\oplus \ZZ = \tilde H_0(\CC)\oplus \ZZ\). \(\square\)

2. Check properties (1)–(5) of chain homotopies. Only (2) and (5) require care.
(1) Chain homotopy is an equivalence relation on the set of chain maps from \(\CC\) to \(\CC'\).

Proof. Let \(\phi, \psi, \varphi \colon \CC\to \CC'\) are chain maps, suppose \(\phi\) and \(\psi\) are chain homotopy, we have \(D\colon C_p \to C_{p+1}'\) satisfying \(\partial' D + D\partial = \psi - \phi\). If \(\psi\) and \(\varphi\) are also chain homotopy, we have \(D'\) satisfying \(\partial' D' + D'\partial = \varphi - \psi\). Therefore, we have \(\partial'(D+D') + (D+D')\partial = \psi - \varphi\), thus \(\phi\) and \(\varphi\) are chain homotopy. This proves the transitiveness.

Clearly, each chain map \(\phi\) is homotopy to itself by taking \(D=0\). In conclusion, chain homotopy is an equivalence relation. \(\square\)

(2) Composition of chain maps induces a well-defined composition operation on chain-homotopy classes.

Proof. Let \(\phi, \psi \colon \CC\to \CC'\) be a chain homotopy, we have \(D\) such that \(\partial' D + D\partial = \psi -\phi\). For a chain map \(\varphi \colon \CC' \to \CC''\), we have

\[ (\psi \circ \varphi)_p - (\phi \circ \varphi)_p = \partial'_{p+1}D_p \varphi_p + D_{p-1} \partial_p \varphi_p = \partial'_{p+1}D_p \varphi_p + D_{p-1} \varphi_{p-1} \partial_p, \]

thus \(D \varphi\) gives a chain homotopy between \(\psi\circ \varphi\) and \(\phi\circ \varphi\).

Similarly, for a chain map \(\varphi \colon \CC'' \to \CC\), \(\varphi_{p+1} D_p\colon C_p''\to C_{p+1}\) gives a chain homotopy between \(\varphi\circ \psi\) and \(\varphi \circ\phi\). \(\square\)

(3) If \(\phi\) and \(\psi\) are chain homotopic, then they induce the same homomorphism in homology.

Proof. For a cycle \(c_p\), we have \(\psi(c_p) - \phi(c_p) = \partial'D(c_p) - D\partial c_p = \partial' D(c_p)\) bounds, thus \(\psi_* - \phi_* = 0\), we have \(\psi_* = \phi_*\). \(\square\)

(4) If \(\phi\) is a chain equivalence, with chain-homotopy inverse \(\phi'\), then \(\phi_*\) and \((\phi')_*\) are homology isomorphisms that are inverse to each other.

Proof. Since \(\phi\) and \(\phi'\) are chain-homotopy inverse to each other, we have \(\phi_* \circ (\phi')_* = (\phi\circ \phi')_* = (\id_{\CC'})_*\), similarly, \((\phi')_* \circ \phi_* = (\id_\CC)_*\), thus \(\phi_*\) and \((\phi')_*\) gives the isomorphism between \(H_p(\CC)\) and \(H_p(\CC')\). \(\square\)

(5) If \(\phi\colon \CC\to \CC'\) and \(\psi\colon \CC'\to\CC''\) are chain equivalences, then \(\psi\circ \phi\) is a chain equivalence.

Proof. Let \(\phi'\colon \CC' \to \CC\) be the chain-homotopy inverse of \(\phi\), thus we have \(D\colon C_p\to C_{p+1}\) satisfying \(\phi'\circ \phi = \id_{\CC} + D\partial + \partial D\).

Similarly, we have \(\psi' \colon \CC''\to \CC'\) and \(D'\colon C'_p \to C'_{p+1}\), such that \(\psi' \circ \psi = \id_{\CC'} + D' \partial' + \partial' D'\). Therefore, we have

\[\begin{align*} (\phi' \circ \psi') \circ (\psi \circ \phi) &= \phi' \circ (\id_{\CC'} + D'\partial' + \partial' D') \circ \phi\\ &= \phi'\circ \phi + \phi' D' \partial' \phi + \phi' \partial' D' \phi\\ &= \id_{\CC} + D\partial + \partial D + \phi' D' \phi \partial + \partial \phi D' \phi\\ &= \id_\CC +(\phi'D'\phi + D) \partial + \partial(\phi'D'\phi + D). \end{align*}\]

Thus \(\phi'D'\phi + D\) gives the chain homotopy between \(\phi'\psi' \psi \phi\) and \(\id_\CC\).
Similarly, \(\psi\phi \phi' \psi'\) is chain homotopy with \(\id_{\CC''}\). This shows that \(\psi \circ \phi\) is a chain equivalence. \(\square\)

6. Prove Theorem 13.6 as follows:
(a) Show \(\phi\) and \(\psi\) are augmentation-preserving chain maps, and show that \(\psi\circ \phi\), equals the identity map of \(\CC(K)\).

Proof. For each \(v_0 < \cdots < v_p\), we have \(\psi \circ \phi([v_0,\dots,v_p]) = \psi((v_0,\dots,v_p)) = [v_0,\dots,v_p]\). Since \(\psi\circ \phi\) acts identically on the generators of \(C_p(K)\), we have \(\psi\circ \phi = \id_{\CC(K)}\). \(\square\)

(b) Define an acyclic carrier from \(\CC'(K)\) to \(\CC'(K)\) that carries both \(\phi\circ \psi\), and the identity map.

Proof. For each basis element \(\sigma_p^\alpha = (v_0,\dots,v_p)\), let \(\Phi(\sigma_p^\alpha)\) be the ordered simplices generated by vertices \(v_0,\dots,v_p\). Being a cone, \(\Phi(\sigma_p^\alpha)\) is augmented by \(\epsilon'\) and is acyclic. And for any term \(\sigma_{p-1}^\beta\) appearing in terms of \(\partial\sigma_p^\alpha\), its vertices is supported by \(v_0,\dots,v_p\), thus we have \(\Phi(\sigma_{p-1}^\beta)\) is a subchain complex of \(\Phi(\sigma_p^\alpha)\).

We can clearly verify that \(\id_{\CC'(K)}\) and \(\phi \circ \psi\) are both carried by \(\Phi\), thus they are chain homotopy. \(\square\)