6. 附录

The author is indebted to 芝麻油 for his great help to and my teaching assistant Jacircen for her careful examination of the solution below. My thanks go also to SHS for offering the opportunity to have several conversations with professor Luzhi when in senior one, without which I would not have taken this course.

4.3

(a) $Z / m \oplus Z / n$ contains an element $(n, m)$ of order $mn$ , so $Z / m \oplus Z / n ≃ Z / mn$ .
(b) $G ≃ Z /2 \oplus Z /9 \oplus Z /4 \oplus Z /9$ .
(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) $H_{1} (X_{n}) = Z^{2 n}$ , $H_{2} (X_{n}) = Z$ .
(b) $H_{1} (Y_{n}) = Z^{n - 1} \oplus Z /2$ , $H_{2} (Y_{n}) = 0$ .

9.6

Using Exercise 39.2, $H_{i} (K, K_{0}) ≃ \tilde{H}_{i} (K / K_{0}) = \tilde{H}_{i} (σ / Bd σ) ≃ \tilde{H}_{i} (S^{2}) = {Z, 0, n = 2 else$ .

10.3

$H_{i} (S; Z /3) = ⎩ ⎨ ⎧ Z /3, Z /3, 0, i = 0 i = 1 i = 2$ . $H_{i} (S; Z /4) = ⎩ ⎨ ⎧ Z /4, Z /4 \oplus Z /2, Z /2, i = 0 i = 1 i = 2$ .

14.1

$f (a) = A^{'}, N^{'}$ , $f (b) = B^{'}$ , $f (c) = C^{'}$ , $f (d) = C^{'}$ , $f (e) = E^{'}$ , $f (f) = E^{'}, F^{'}$ , $f (g) = F^{'}, G^{'}$ , $f (h) = G^{'}, H^{'}$ , $f (i) = I^{'}$ , $f (j) = J^{'}$ , $f (k) = K^{'}, L^{'}$ , $f (l) = L^{'}$ , $f (m) = L^{'}, M^{'}$ , $f (n) = N^{'}$ , $f (o) = O^{'}$ , altogether $2^{6} = 64$ simplicial approximations.

21.2

(a) Note that the ball $∣ z ∣ \leq c ≃$ an interior point $0$ , and so is image $h (∣ z ∣ \leq c) ≃ h (0)$ in $R^{2} - 0$ . In particualr, the boundary $h (S_{c})$ collapses into a point $h (0)$ . It follows that $h_{*} {S_{c}} = h_{*} {0} = 0$ .

(b) Set $F (z, t) = z^{n} + t (a_{n - 1} z^{n - 1} + \dots + a_{0})$ . The image of $F$ never meets $0$ if $c$ is sufficiently large since $∣ F (z, t) ∣ \geq c^{n} - t (∣ a_{n - 1} ∣ c^{n - 1} + \dots + ∣ a_{0} ∣) > 0$ . This is a homotopy between $F (-, 1) = h$ and $F (-, 0) = k$ .

(c) $h_{*} = k_{*} : H_{1} (S_{c}) \to H_{1} (R^{2} - 0)$ by (b), yet $h_{*} = 0, k_{*} = n$ .

22.4

(a) $F (-, t) : M \to M$ for a fixed $t \in (- ϵ, ϵ)$ is a map with no fixed point, and $≃ id$ as $-$ varies from $t$ to $0$ . Then $χ (M) = Λ (id) = 0$ .

(b) Exactly those with $χ (M) = 0$ , i.e. the torus or the Klein bottle.

24.7

The long exact sequence we shall use is

(a)	For $∣ K ∣ = P^{2}$ and $0 \to Z \to Z \to Z /2 \to 0$ , we haveFor $∣ K ∣ = P^{2}$ and $0 \to Z /2 \to Z /4 \to Z /2 \to 0$ , we have
(b)	For $∣ K ∣ =$ Klein and $0 \to Z \to Z \to Z /2 \to 0$ , we haveFor $∣ K ∣ =$ Klein and $0 \to Z /2 \to Z /4 \to Z /2 \to 0$ , we have

25.3

$S^{n + 1}$ is the suspension of $S^{n}$ , and let $S f : (w * S^{n}) / S^{n} \to (w * S^{n}) / S^{n}$ induced by $f$ be the suspension of $f$ . The naturality of the boundary maps in the long exact sequence of the pair $(w * S^{n}, S^{n})$ gives the commutative diagram, so $de g S f = de g f$ . $\tilde{H}_{n + 1} (S^{n + 1}) \tilde{H}_{n} (S^{n}) \tilde{H}_{n + 1} (S^{n + 1}) \tilde{H}_{n} (S^{n}) ≃ \partial S f_{*} f_{*} ≃ \partial$ The conclusion follows from induction and $k : S^{1} \to S^{1}, k (z) = z^{n}$ has degree $n$ .

26.2

Let us take a look at a grotesque diagram below. $H_{p} (X) H_{p} (X, A) H_{p - 1} (A, B) H_{p - 2} (B) H_{p} (A) H_{p} (X, B) H_{p - 1} (A) H_{p - 1} (X, B) H_{p} (A, B) H_{p - 1} (B) H_{p - 1} (X) i_{*} j_{*} β \partial_{*} \partial_{*} π k_{*} i_{*} η \partial_{*} i_{*} k_{*} \partial_{*} \partial_{*} π j_{*} i_{*} j_{*}$ The three sequences whose maps are of the same color are exact. From the exactness of $H_{p} (A) ⟶ i_{*} H_{p} (A) ⟶ π_{*} H_{p} (A, A) ⟶ \partial_{*} H_{p - 1} (A)$ we deduce that $H_{p} (A, A) = 0$ , and due to the commutativity (as all four maps are mere inclusions) of $H_{p} (X, B) H_{p} (A, B) H_{p} (X, A) H_{p} (A, A) η π η^{'} π^{'}$ we get $η \circ π = 0$ . Then apply the braid lemma in Exercise 1(b), we obtain the exactness of the sequence with Greek alphabets above the arrows in the above diagram.

26.3

[Hatcher] Section 2.2 Exercise 38, p.159.

29.1

(b) By (a), $H_{p} (tsc) = {Z^{2}, 0, p = 0 p > 0$ .

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. $\partial D_{Δ_{p}} i_{p} = j \leq i \sum (- 1)^{i} (- 1)^{j} l ((ϵ_{0}, 0), \dots, (ϵ_{j}, 0), \dots, (ϵ_{i}, 0), (ϵ_{i}, 1), \dots, (ϵ_{p}, 1)) + j \geq i \sum (- 1)^{i} (- 1)^{j + 1} l ((ϵ_{0}, 0), \dots, (ϵ_{i}, 0), (ϵ_{i}, 1), \dots, (ϵ_{j}, 1), \dots, (ϵ_{p}, 1)) .$ The terms with $i = j$ in the two sums cancel except for $l ((ϵ_{0}, 0), (ϵ_{0}, 1), \dots, (ϵ_{p}, 1))$ , which is $j \circ i_{p} = j_{#} (i_{p})$ , and $- l ((ϵ_{0}, 0), \dots, (ϵ_{p}, 0), (ϵ_{p}, 1))$ , which is $- i \circ i_{p} = - i_{#} (i_{p})$ . The terms with $i \neq = j$ are exactly $- D_{Δ_{p}} \partial i_{p}$ since $D_{Δ_{p}} \partial i_{p} = i < j \sum (- 1)^{i} (- 1)^{j} l ((ϵ_{0}, 0), \dots, (ϵ_{i}, 0), (ϵ_{i}, 1), \dots, (ϵ_{j}, 1), \dots, (ϵ_{p}, 1)) + j > i \sum (- 1)^{i - 1} (- 1)^{j} l ((ϵ_{0}, 0), \dots, (ϵ_{j}, 0), \dots, (ϵ_{i}, 0), (ϵ_{i}, 1), \dots, (ϵ_{p}, 1)) .$

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 $H_{1} (X_{1}) \oplus H_{1} (X_{2}) ⟶ H_{1} (X) ⟶ \partial H_{0} (X_{1} \cap X_{2}) ⟶ i H_{0} (X_{1}) \oplus H_{0} (X_{2})$ All path components of $X_{1}$ and $X_{2}$ are contractible, so $H_{1} (X_{1}) \oplus H_{1} (X_{2}) = 0$ , and consequently $H_{1} (X) ≃ im \partial = ker i = 0$ . And $H_{0} (X) = Z$ 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 $H_{p} (S^{n}) = {Z, 0, p = 0, n else$ .

35.4

(a) As $\overline{St} v$ is contractible and all of its homology groups vanish except for $p = 0$ , for $p > 0$ $H_{p} (Lk v) ≃ H_{p} (\overline{St} v - v) ≃ H_{p + 1} (\overline{St} v, \overline{St} v - v) = {Z, 0, p = n - 1 else$ when $x \in Int M$ or $0$ when $x \in Bd M$ .

(b) Suppose that there is a simplex $σ$ of dimension less than $m$ that is not a face of any simplex of dimension $m$ . Then $Lk \overset{σ}{^} = Bd σ$ is a sphere of dimension less than $m - 1$ , contradicting (a) for $v = \overset{σ}{^}$ and $K \cup sd σ$ 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 $\overset{s}{^}$ is an interior point of $K$ . Therefore $s$ is a face of precisely two $m$ -simplices iff $\overset{s}{^}$ is an interior point for any neighborhood of $\overset{s}{^}$ in $K$ , iff $h (s) \in Int M$ ; on the contrary $s$ is a face of precisely one $m$ -simplex iff $h (s) \in Bd M$ .

so $s$ is a face of precisely two $m$ -simplices iff $h (s) \in Bd M$ .

35.5

$S^{3} ≃ Bd (B^{2} \times B^{2}) = (B^{2} \times Bd B^{2}) \cup (Bd B^{2} \times B^{2}) ≃ T_{1} \cup T_{2}$ , and $T_{1} \cap T_{2} ≃ Bd B^{2} \times Bd B^{2} = S^{1} \times S^{1} =$ a hollow torus $T$ , so

38.1

[Rotman] Lemma 8.18, Theorem 8.19, pp.201–2.

Lemma. For a cell $e \in X$ , the closure $\overset{e}{ˉ}$ 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

\overset{e}{ˉ} - e = g (Bd B) \subset X^{p - 1}

. By “closure–finiteness”,

\overset{e}{ˉ}

meets only finitely many cells other than

e

, say,

e_{1}, \dots, e_{m}

, and we have just seen that

dim e_{i} \leq p - 1

for all

i

. By induction, there is a finite CW subcomplex

X_{i}

containing

\overset{e}{ˉ}_{i}

, for

i = 1, \dots, m

, and each

X_{i}

is closed. But

\overset{e}{ˉ} \subset e \cup X_{1} \cup \dots \cup X_{m}

, so that this union of finitely many cells is closed and hence is a finite CW subcomplex.

$□$

(a) For each cell $e$ with $A \cap e \neq = \emptyset$ , 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 $\overset{e}{ˉ}$ . Therefore $E \cap \overset{e}{ˉ} \subset E \cap X_{e}$ is a finite set and hence is closed in $\overset{e}{ˉ}$ . 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 $\overset{e}{ˉ}_{i} \subset X_{i}$ for $i = 1, \dots, m$ . It follows that $A$ lies in the finite CW subcomplex $1 \leq i \leq m ⋃ X_{i}$ .

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 ${f_{α} : B_{α} \to X}_{α}$ be the family of the characteristic maps of $X$ , ${g_{β} : B_{β} \to Y}_{β}$ be the family of the characteristic maps of $X$ . Then ${(f_{α}, g_{β}) : B_{α} \times B_{β} \to X \times Y}_{α, β}$ 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 $B_{α} \times B_{β}$ . 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 ${(e_{0}, e_{1}, \dots, e_{n}) ∣ each e_{i} is a cell, e_{0} ⊊ e_{1} ⊊ \dots ⊊ e_{n}}$ . Let $∣ T (X) ∣$ be the geometric realization of this complex. We define a homeomorphism $Ψ : ∣ T (X) ∣ \to X$ such that if $e$ is a cell of $X$ , $Ψ^{- 1} (e)$ is a subcomplex of $∣ T (X) ∣$ .

The map $Ψ$ is constructed by first noticing that $T (X) = n ⋃ T (X^{n})$ and inductively constructing $Ψ^{n} : ∣ T (X^{n}) ∣ \to X^{n}$ so that $Ψ^{n}$ 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 $Ψ^{n - 1}$ is defined and $e^{n}$ is an $n$ -cell, then $Bd e^{n}$ is a subcomplex of $X^{n - 1}$ , and $(Ψ^{n - 1})^{- 1} (Bd e^{n})$ is a subcomplex of $∣ T (X^{n}) ∣$ which is homeomorphic to the $(n - 1)$ -sphere. The usual conical construction enables us to extend $Ψ^{n - 1}$ over $∣ T (X^{n}) ∣$ .

Geometrically, we are performing 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 diagram $H_{p} (X, A) H_{p} (X, V) H_{p} (X - A, V - A) H_{p} (X / A, A / A) H_{p} (X / A, V / A) H_{p} (X / A - A / A, V / A - A / A) q_{*} q_{*} q_{*}$ The 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 $(V, A) ≃ (A, A)$ , 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 $q_{*}$ is an isomorphism.

39.3

$D_{p} (A)$ is a free abelian group with a basis ${(f_{α})_{*} (γ) : f_{α}$ 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. $H_{p} (D (A)) H_{p} (D (X)) H_{p} (D (X, A)) H_{p - 1} (D (A)) H_{p - 1} (D (X)) H_{p} (A) H_{p} (X) H_{p} (X, A) H_{p - 1} (A) H_{p - 1} (X) ≃ ≃ ≃ ≃$ (b) Let $X_{p} = X^{p} \cup A$ for all integers $p$ . 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 $H_{i} (X_{p}, X_{p - 1}) ≃ H_{i} (X, A)$ (for a detailed proof, see [Jiang] Section 3.2, 3.3, pp.99–104).

Plus, the isomorphism between $D (X, A) ≃ α ⨁ H_{p} (B_{α}^{p}, S_{α}^{p - 1})$ and $H_{p} (X_{p}, X_{p - 1})$ is given by $α ⨁ H_{p} (B^{p}, S^{p - 1}) ⟶ α ⨁ (f_{α})_{*} H_{p} (X_{p}, X_{p - 1}), H_{p} (X_{p}, X_{p - 1}) ⟶ α ⨁ (q_{α})_{*} α ⨁ H_{p} (B^{p}, S^{p - 1}),$ where the index $α$ is taken over all $p$ -cells in $X - A$ , $f_{α}$ is the characteristic map for $e_{α}^{p}$ , and $q_{α}$ is the quotient map collapsing $X^{p} - e_{α}^{p}$ to a point.

39.4

(a) Let $γ$ be the map induced by the following diagram $(E_{+}^{n}, S^{n - 1}) (S^{n}, E_{-}^{n}) (σ, Bd σ) (S^{n}, x_{0}) ≃ g h γ$ where $E_{-}^{n}$ 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 $H_{n} (E_{+}^{n}, S^{n - 1}) ≃ H_{n} (S^{n} - U, E_{-}^{n} - U) ≃ H_{n} (S^{n}, E_{-}^{n})$ 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 $ϕ$ is multiplication by $d$ , then it is the homomorphism induced by $γ$ composite a map from $S^{n}$ to $S^{n}$ of degree $d$ .

(b) This is because the inclusions induce an isomorphism $\oplus (i_{α})_{*} : α ⨁ H_{n} (S_{α}^{n}, p) \to H_{n} (X, p)$ .

(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 $k_{p - 1} : B_{p - 1} \to C_{p}$ such that $\partial \circ k_{p - 1} = id_{B_{p - 1}}$ , and therefore $C_{p} = k_{p - 1} B_{p - 1} \oplus Z_{p}$ . $B_{p - 1} 0 Z_{p} C_{p} B_{p - 1} 0 k_{p - 1} \partial$ 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. $H_{0} (C) 0 U_{1} C_{0} H_{0} (C) 0 k_{- 1} \partial / \partial U_{1}$ (d) By (c), it suffices to construct a CW complex $X$ whose cellular chain complex is isomorphic to $\partial : U_{n} \to Z_{n - 1} \to 0 \to \dots \to 0 \to Z$ for each $n$ (and $\partial U_{1}$ in place of $Z_{- 1}$ for $n = 0$ ) where the final $Z$ is generated by a point in $A$ , since $\partial = 0$ on $Z_{n}$ and $im \partial \subset ker \partial$ .

Both $U_{n}$ and $Z_{n - 1}$ are free abelian groups, bases denoted ${x_{α}}$ and ${y_{β}}$ , and write $\partial x_{α} = β \sum d_{β α} y_{β}$ . Let $X^{n} = α ⋁ S_{α}^{n}$ . We construct $X$ from $X^{n}$ by attaching cells $e_{β}^{n + 1}$ via maps $f_{β} : S^{n} \to X^{n}$ such that the projection onto the summand $S_{α}^{n}$ has degree $d_{β α}$ . Plus, generate $A$ by other points if $∣ A ∣ > 1$ .

43.2

(a) Let us take $σ_{2}^{*}, σ_{1}^{*} - σ_{2}^{*}, σ_{2}^{*} - σ_{3}^{*}, σ_{3}^{*} - σ_{4}^{*}, σ_{4}^{*} - σ_{5}^{*}, σ_{5}^{*} - σ_{6}^{*}$ as a basis for $Z^{2} (C, E)$ , and $e_{7}^{*}, e_{1}^{*}, e_{2}^{*}, e_{3}^{*}, e_{4}^{*}, e_{5}^{*}, e_{1}^{*} + e_{2}^{*} + e_{3}^{*} + e_{4}^{*} + e_{5}^{*} + e_{6}^{*}, e_{2}^{*} + e_{3}^{*} - e_{7}^{*} + e_{8}^{*}, e_{4}^{*} + e_{5}^{*} - e_{8}^{*} + e_{9}^{*}$ as a basis for $C^{1} (C, E)$ . We see that $δ e_{i}^{*} = σ_{i}^{*} - σ_{i + 1}^{*}$ for $i = 1, \dots, 5$ ; and $δ e_{6}^{*} = σ_{6}^{*} - σ_{1}^{*}, δ e_{7}^{*} = σ_{2}^{*}, δ e_{8}^{*} = σ_{4}^{*}, δ e_{9}^{*} = σ_{6}^{*} .$ Thus $δ$ is surjective, i.e. $H^{2} (C, E) = 0$ .Note that $δ (e_{1}^{*} + \dots + e_{6}^{*}) = 0$ , and $δ (e_{2}^{*} + e_{3}^{*} - e_{7}^{*} + e_{8}^{*}) = 0, δ (e_{4}^{*} + e_{5}^{*} - e_{8}^{*} + e_{9}^{*}) = 0.$ That is to say, $e_{1}^{*} + \dots + e_{6}^{*}, e_{2}^{*} + e_{3}^{*} - e_{7}^{*} + e_{8}^{*}, e_{4}^{*} + e_{5}^{*} - e_{8}^{*} + e_{9}^{*}$ constitute a basis for $1$ -cocycles; but they are $1$ -coboundaries as well, as $- δ (v_{3}^{*} + v_{4}^{*} + v_{5}^{*}) = e_{1}^{*} + \dots + e_{6}^{*}, - δ v_{4}^{*} = e_{2}^{*} + e_{3}^{*} - e_{7}^{*} + e_{8}^{*}, - δ v_{5}^{*} = e_{4}^{*} + e_{5}^{*} - e_{8}^{*} + e_{9}^{*},$ and therefore $H^{1} (C, E) = 0$ . Finally, $H^{0} (C, E) = 0$ .

(b) The exact sequence $0 H^{0} (C, E) H^{0} (C) H^{0} (E) H^{1} (C, E) H^{1} (C) H^{1} (E) H^{2} (C, E) H^{2} (C) H^{2} (E) 0 ≃ ≃ ≃$ yields $H^{i} (C) ≃ H^{i} (E) = {Z, 0, i = 0, 1 i = 2$ .

43.3

(a), (b), (f).

44.3

This is inferred by a general fact that if a chain map between free chain complexes $ϕ : C \to C^{'}$ induces isomorphisms $ϕ_{*} : H_{p} (C) \to H_{p} (C^{'})$ on homology groups for all $p$ , then it also induces isomorphisms $ϕ^{*} : H^{p} (C^{'}; G) \to H^{p} (C; G)$ on cohomology groups with any coefficient group $G$ for all $p$ . The proof of the fact is an application of the Five–lemma to the diagram below. $0 Ext (H_{p - 1} (C), G) H^{p} (C; G) Hom (H_{p} (C), G) 00 Ext (H_{p - 1} (C^{'}), G) H^{p} (C^{'}; G) Hom (H_{p} (C^{'}), G) 0 ≃ ≃$

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.

名字空间

视图

6. 附录

参考文献