Nakahara 第3章 · 来源与作者路径

逐字来源、作者主张与候选译文分别保留。工程核验不批准 OCR、译文或数学正确性。

概念与作者路径

§3.1 阿贝尔群约定

本章所有群采用阿贝尔群的加法记号。

条件与约定：Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。

符号：$+$；$0$

逐字作者来源与版本

The mathematical structures underlying homology groups are finitely generated Abelian groups. Throughout this chapter, the group operation is denoted by  $+$  since all the groups considered here are Abelian (commutative). The unit element is denoted by 0.

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§3.1.1 同态 (3.1)

阿贝尔群间的映射对任意两元素保持加法。

条件与约定：Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。

符号：$f:G_{1}o{G}_2$

逐字作者来源与版本

Let  $G_{1}$  and  $G_{2}$  be Abelian groups. A map  $f: G_{1} \to G_{2}$  is said to be a homomorphism if

$$
f (x + y) = f (x) + f (y) \tag {3.1}
$$

for any  $x, y \in G_1$ . If  $f$  is also a bijection,  $f$  is called an isomorphism. If there exists an isomorphism  $f: G_1 \to G_2$ ,  $G_1$  is said to be isomorphic to  $G_2$ , denoted by  $G_1 \cong G_2$ . For example, a map  $f: \mathbb{Z} \to \mathbb{Z}_2 = \{0, 1\}$  defined by

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§3.1.1 子群

子集在母群原运算下构成群时称为子群。

条件与约定：Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。

符号：$H\subset G$

逐字作者来源与版本

A subset  $H \subset G$  is a subgroup if it is a group with respect to the group operation of  $G$ . For example,

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§3.1.1 商群 (3.2)–(3.3)

以模子群的等价类定义商，并验证代表元选择不影响加法。

条件与约定：Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。 本章每个子群均正规，不能省去一般群商构造中的正规性条件。

符号：$G/H$；$[x]+[y]=[x+y]$

逐字作者来源与版本

Let  $H$  be a subgroup of  $G$ . We say  $x, y \in G$  are equivalent if

$$
x - y \in H \tag {3.2}
$$

and write  $x \sim y$ . Clearly  $\sim$  is an equivalence relation. The equivalence class to which  $x$  belongs is denoted by  $[x]$ . Let  $G / H$  be the quotient space. The group operation  $+$  in  $G$  naturally induces the group operation  $+$  in  $G / H$  by

$$
[ x ] + [ y ] = [ x + y ]. \tag {3.3}
$$

Note that  $+$  on the LHS is an operation in  $G / H$  while  $+$  on the RHS is that in  $G$ . The operation in  $G / H$  should be independent of the choice of representatives. In fact, if  $[x'] = [x]$ ,  $[y'] = [y]$ , then  $x - x' = h$ ,  $y - y' = g$  for some  $h, g \in H$  and we find that

$$
x ^ {\prime} + y ^ {\prime} = x + y - (h + g) \in [ x + y ]
$$

Furthermore,  $G / H$  becomes a group with this operation, since  $H$  is always a normal subgroup of  $G$ ; see example 2.6. The unit element of  $G / H$  is  $[0] = [h]$ ,

$h \in H$ . If  $H = G$ ,  $0 - x \in G$  for any  $x \in G$  and  $G / G$  has just one element [0]. If  $H = \{0\}$ ,  $G / H$  is  $G$  itself since  $x - y = 0$  if and only if  $x = y$ .

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后文使用：定义3.5：同调群及作者的良定义说明。

引理3.1 核与像为子群

群同态的核与像是相应群的子群。

条件与约定：Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。

逐字作者来源与版本

Lemma 3.1. Let  $f: G_1 \to G_2$  be a homomorphism. Then

(a)  $\ker f = \{x|x\in G_1,f(x) = 0\}$  is a subgroup of  $G_{1}$
(b)  $\operatorname {im}f = \{x|x\in f(G_1)\subset G_2\}$  is a subgroup of  $G_{2}$

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后文使用：定理3.1 的证明。

后文使用：循环群和边界群的子群资格。

定理3.1 同态基本定理

作者给出 G1/ker f 与 im f 的同构。

条件与约定：Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。

逐字作者来源与版本

Theorem 3.1. (Fundamental theorem of homomorphism) Let  $f: G_1 \to G_2$  be a homomorphism. Then

$$
G _ {1} / \ker f \cong \operatorname {i m} f. \tag {3.6}
$$

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后文使用：例3.2 对定理3.1的应用。

定理3.1 的证明

先据引理3.1确定两边为群，再构造并验证同构。

条件与约定：Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。

逐字作者来源与版本

Proof. Both sides are groups according to lemma 3.1. Define a map  $\varphi : G_1 / \ker f \to \operatorname{im} f$  by  $\varphi([x]) = f(x)$ . This map is well defined since for  $x' \in [x]$ , there exists  $h \in \ker f$  such that  $x' = x + h$  and  $f(x') = f(x + h) = f(x) + f(h) = f(x)$ . Now we show that  $\varphi$  is an isomorphism. First,  $\varphi$  is a homomorphism,

$$
\begin{array}{l} \varphi ([ x ] + [ y ]) = \varphi ([ x + y ]) = f (x + y) \\ = f (x) + f (y) = \varphi ([ x ]) + \varphi ([ y ]). \\ \end{array}
$$

Second,  $\varphi$  is one to one: if  $\varphi([x]) = \varphi([y])$ , then  $f(x) = f(y)$  or  $f(x) - f(y) = f(x - y) = 0$ . This shows that  $x - y \in \ker f$  and  $[x] = [y]$ . Finally,  $\varphi$  is onto: if  $y \in \operatorname{im} f$ , there exists  $x \in G_1$  such that  $f(x) = y = \varphi([x])$ .

PDF 文件页：114；OCR 未因此获得逐页核验。

使用前置陈述：引理3.1 核与像为子群。作者有明确编号回指。

证据：作者编号回指

Proof. Both sides are groups according to lemma 3.1. Define a map  $\varphi : G_1 / \ker f \to \operatorname{im} f$  by  $\varphi([x]) = f(x)$ . This map is well defined since for  $x' \in [x]$ , there exists  $h \in \ker f$  such that  $x' = x + h$  and  $f(x') = f(x + h) = f(x) + f(h) = f(x)$ . Now we show that  $\varphi$  is an isomorphism. First,  $\varphi$  is a homomorphism,

例3.2 对定理3.1的应用

对整数到二元循环群的映射使用同态基本定理。

条件与约定：Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。

逐字作者来源与版本

Example 3.2. Let  $f: \mathbb{Z} \to \mathbb{Z}_2$  be defined by  $f(2n) = 0$  and  $f(2n + 1) = 1$ . Then  $\ker f = 2\mathbb{Z}$  and  $\operatorname{im} f = \mathbb{Z}_2$  are groups. Theorem 3.1 states that  $\mathbb{Z}/2\mathbb{Z} \cong \mathbb{Z}_2$ , in agreement with example 3.1.

PDF 文件页：114；OCR 未因此获得逐页核验。

使用前置陈述：定理3.1 同态基本定理。作者有明确编号回指。

证据：作者编号回指

Example 3.2. Let  $f: \mathbb{Z} \to \mathbb{Z}_2$  be defined by  $f(2n) = 0$  and  $f(2n + 1) = 1$ . Then  $\ker f = 2\mathbb{Z}$  and  $\operatorname{im} f = \mathbb{Z}_2$  are groups. Theorem 3.1 states that  $\mathbb{Z}/2\mathbb{Z} \cong \mathbb{Z}_2$ , in agreement with example 3.1.

定义3.2：整数系数链群

有向 r 单形生成自由阿贝尔链群，链的系数为整数。

条件与约定：本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。

符号：$C_r(K)$

逐字作者来源与版本

Definition 3.2. The  $r$ -chain group  $C_r(K)$  of a simplicial complex  $K$  is a free Abelian group generated by the oriented  $r$ -simplexes of  $K$ . If  $r > \dim K$ ,  $C_r(K)$  is defined to be 0. An element of  $C_r(K)$  is called an  $r$ -chain.

Let there be  $I_r$ $r$ -simplexes in  $K$ . We denote each of them by  $\sigma_{r,i}$  ( $1 \leq i \leq I_r$ ). Then  $c \in C_r(K)$  is expressed as

$$
c = \sum_ {i = 1} ^ {I _ {r}} c _ {i} \sigma_ {r, i} \quad c _ {i} \in \mathbb {Z}. \tag {3.15}
$$

{"image_caption": [], "image_footnote": [], "img_path": "images/ddbc60f39a21b01109470e0abf58cca1ab547d5c55706fcee0682d813b959a74.jpg"}

Figure 3.7. (a) An oriented 1-simplex with a fictitious boundary  $p_1$ . (b) A simplicial complex without a boundary.

The integers  $c_{i}$  are called the coefficients of  $c$ . The group structure is given as follows. The addition of two  $r$ -chains,  $c = \sum_{i} c_{i} \sigma_{r,i}$  and  $c' = \sum_{i} c_{i}' \sigma_{r,i}$ , is

$$
c + c ^ {\prime} = \sum_ {i} \left(c _ {i} + c _ {i} ^ {\prime}\right) \sigma_ {r, i}. \tag {3.16}
$$

The unit element is  $0 = \sum_{i} 0 \cdot \sigma_{r,i}$ , while the inverse element of  $c$  is  $-c = \sum_{i} (-c_i) \sigma_{r,i}$ . [Remark: An oppositely oriented  $r$ -simplex  $-\sigma_r$  is identified with  $(-1) \sigma_r \in C_r(K)$ .] Thus,  $C_r(K)$  is a free Abelian group of rank  $I_r$ ,

$$
C _ {r} (K) \cong \underbrace {\mathbb {Z} \oplus \mathbb {Z} \oplus \cdots \oplus \mathbb {Z}} _ {I _ {r}}. \tag {3.17}
$$

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后文使用：定义3.5：同调群及作者的良定义说明。

后文使用：边界同态：式(3.20)–(3.22)。

边界同态：式(3.20)–(3.22)

按有向顶点的交错和定义边界，并线性延拓为相邻链群之间的同态。

条件与约定：本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。

符号：$$\begin{gathered} partia{l}_r:C_r(K) \\ to{C}_{r-1}(K) \end{gathered}$$

逐字作者来源与版本

Let  $\sigma_r(p_0\ldots p_r)$  ( $r > 0$ ) be an oriented  $r$ -simplex. The boundary  $\partial_r\sigma_r$  of  $\sigma_r$  is an  $(r - 1)$ -chain defined by

$$
\partial_ {r} \sigma_ {r} \equiv \sum_ {i = 0} ^ {r} (- 1) ^ {i} \left(p _ {0} p _ {1} \dots \hat {p} _ {i} \dots p _ {r}\right) \tag {3.20}
$$

where the point  $p_i$  under  $\hat{\mathbf{\mu}}$  is omitted. For example,

$$
\begin{array}{l} \partial_ {2} \left(p _ {0} p _ {1} p _ {2}\right) = \left(p _ {1} p _ {2}\right) - \left(p _ {0} p _ {2}\right) + \left(p _ {0} p _ {1}\right) \\ \partial_ {3} \left(p _ {0} p _ {1} p _ {2} p _ {3}\right) = \left(p _ {1} p _ {2} p _ {3}\right) - \left(p _ {0} p _ {2} p _ {3}\right) + \left(p _ {0} p _ {1} p _ {3}\right) - \left(p _ {0} p _ {1} p _ {2}\right). \\ \end{array}
$$

We formally define  $\partial_0\sigma_0 = 0$  for  $r = 0$ .

The operator  $\partial_r$  acts linearly on an element  $c = \sum_{i} c_i \sigma_{r,i}$  of  $C_r(K)$ ,

$$
\partial_ {r} c = \sum_ {i} c _ {i} \partial_ {r} \sigma_ {r, i}. \tag {3.21}
$$

The RHS of (3.21) is an element of  $C_{r-1}(K)$ . Accordingly,  $\partial_r$  defines a map

$$
\partial_ {r}: C _ {r} (K) \rightarrow C _ {r - 1} (K). \tag {3.22}
$$

$\partial_r$  is called the boundary operator. It is easy to see that the boundary operator is a homomorphism.

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后文使用：定义3.3：循环群是边界同态的核。

后文使用：定义3.4：边界群是前一边界同态的像。

使用前置陈述：定义3.2：整数系数链群。作者将边界线性延拓到链，因此映射的定义域、值域取自定义3.2的链群。

证据：作者承接

The RHS of (3.21) is an element of  $C_{r-1}(K)$ . Accordingly,  $\partial_r$  defines a map

后文使用：引理3.3 连续两次取边界为零。

定义3.3：循环群是边界同态的核

r 循环的边界为零，循环群是边界同态的核。

条件与约定：本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。

符号：$$\begin{gathered} Z_r(K)= \\ ker \\ partia{l}_r \end{gathered}$$

逐字作者来源与版本

Definition 3.3. If  $c \in C_r(K)$  satisfies

$$
\partial_ {r} c = 0 \tag {3.24}
$$

$c$  is called an  $r$ -cycle. The set of  $r$ -cycles  $Z_{r}(K)$  is a subgroup of  $C_{r}(K)$  and is called the  $r$ -cycle group. Note that  $Z_{r}(K) = \ker \partial_{r}$ . [Remark: If  $r = 0$ ,  $\partial_0 c$  vanishes identically and  $Z_{0}(K) = C_{0}(K)$ , see (3.23).]

PDF 文件页：124；OCR 未因此获得逐页核验。

后文使用：定义3.5：同调群及作者的良定义说明。

后文使用：循环群和边界群的子群资格。

使用前置陈述：边界同态：式(3.20)–(3.22)。循环条件和核需要此前定义的边界同态。

证据：构造使用（待独立复核）

$c$  is called an  $r$ -cycle. The set of  $r$ -cycles  $Z_{r}(K)$  is a subgroup of  $C_{r}(K)$  and is called the  $r$ -cycle group. Note that  $Z_{r}(K) = \ker \partial_{r}$ . [Remark: If  $r = 0$ ,  $\partial_0 c$  vanishes identically and  $Z_{0}(K) = C_{0}(K)$ , see (3.23).]

定义3.4：边界群是前一边界同态的像

r 边界来自 r+1 链的边界，边界群是其像。

条件与约定：本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。

符号：$$\begin{gathered} B_r(K)= \\ operatornameim \\ partia{l}_{r+1} \end{gathered}$$

逐字作者来源与版本

Definition 3.4. Let  $K$  be an  $n$ -dimensional simplicial complex and let  $c \in C_r(K)$ . If there exists an element  $d \in C_{r + 1}(K)$  such that

$$
c = \partial_ {r + 1} d \tag {3.25}
$$

then  $c$  is called an  $r$ -boundary. The set of  $r$ -boundaries  $B_r(K)$  is a subgroup of  $C_r(K)$  and is called the  $r$ -boundary group. Note that  $B_r(K) = \operatorname{im} \partial_{r+1}$ . [Remark:  $B_n(K)$  is defined to be 0.]

PDF 文件页：124；OCR 未因此获得逐页核验。

后文使用：定义3.5：同调群及作者的良定义说明。

后文使用：循环群和边界群的子群资格。

使用前置陈述：边界同态：式(3.20)–(3.22)。边界及像需要此前从 r+1 链到 r 链的边界同态。

证据：构造使用（待独立复核）

then  $c$  is called an  $r$ -boundary. The set of  $r$ -boundaries  $B_r(K)$  is a subgroup of  $C_r(K)$  and is called the  $r$ -boundary group. Note that  $B_r(K) = \operatorname{im} \partial_{r+1}$ . [Remark:  $B_n(K)$  is defined to be 0.]

循环群和边界群的子群资格

作者明确从引理3.1得到 Zr 与 Br 是链群的子群。

条件与约定：已引入边界同态，尚不把 Br 包含于 Zr 当作这里已证明。

逐字作者来源与版本

From lemma 3.1, it follows that  $Z_{r}(K)$  and  $B_{r}(K)$  are subgroups of  $C_{r}(K)$ . We now prove an important relation between  $Z_{r}(K)$  and  $B_{r}(K)$ , which is crucial in the definition of homology groups.

PDF 文件页：124；OCR 未因此获得逐页核验。

使用前置陈述：引理3.1 核与像为子群。作者有明确编号回指。

证据：作者编号回指

From lemma 3.1, it follows that  $Z_{r}(K)$  and  $B_{r}(K)$  are subgroups of  $C_{r}(K)$ . We now prove an important relation between  $Z_{r}(K)$  and  $B_{r}(K)$ , which is crucial in the definition of homology groups.

后文使用：定义3.5：同调群及作者的良定义说明。

使用前置陈述：定义3.3：循环群是边界同态的核。引理3.1应用到核时，核正是定义3.3的循环群。

证据：构造使用（待独立复核）

From lemma 3.1, it follows that  $Z_{r}(K)$  and  $B_{r}(K)$  are subgroups of  $C_{r}(K)$ . We now prove an important relation between  $Z_{r}(K)$  and  $B_{r}(K)$ , which is crucial in the definition of homology groups.

使用前置陈述：定义3.4：边界群是前一边界同态的像。引理3.1应用到像时，像正是定义3.4的边界群。

证据：构造使用（待独立复核）

From lemma 3.1, it follows that  $Z_{r}(K)$  and  $B_{r}(K)$  are subgroups of  $C_{r}(K)$ . We now prove an important relation between  $Z_{r}(K)$  and  $B_{r}(K)$ , which is crucial in the definition of homology groups.

引理3.3 连续两次取边界为零

边界算子的复合为零。

条件与约定：r 链与边界算子按本书有向单形约定。

逐字作者来源与版本

Lemma 3.3. The composite map  $\partial_r\circ \partial_{r + 1}:C_{r + 1}(K)\to C_{r - 1}(K)$  is a zero map; that is,  $\partial_r(\partial_{r + 1}c) = 0$  for any  $c\in C_{r + 1}(K)$ .

PDF 文件页：124；OCR 未因此获得逐页核验。

后文使用：定理3.3 边界群包含于循环群。

使用前置陈述：边界同态：式(3.20)–(3.22)。连续取边界的零映射结论使用此前的边界算子；这条边不替代引理3.3的证明。

证据：构造使用（待独立复核）

Lemma 3.3. The composite map  $\partial_r\circ \partial_{r + 1}:C_{r + 1}(K)\to C_{r - 1}(K)$  is a zero map; that is,  $\partial_r(\partial_{r + 1}c) = 0$  for any  $c\in C_{r + 1}(K)$ .

定理3.3 边界群包含于循环群

证明明确使用引理3.3。

条件与约定：这一步为同调商群提供子群包含关系。

逐字作者来源与版本

Theorem 3.3. Let  $Z_{r}(K)$  and  $B_{r}(K)$  be the  $r$ -cycle group and the  $r$ -boundary group of  $C_{r}(K)$ , then

$$
B _ {r} (K) \subset Z _ {r} (K) \quad (\subset C _ {r} (K)). \tag {3.27}
$$

Proof. This is obvious from lemma 3.3. Any element  $c$  of  $B_r(K)$  is written as  $c = \partial_{r+1}d$  for some  $d \in C_{r+1}(K)$ . Then we find  $\partial_r c = \partial_r(\partial_{r+1}d) = 0$ , that is,  $c \in Z_r(K)$ . This implies  $Z_r(K) \supset B_r(K)$ .

PDF 文件页：125；OCR 未因此获得逐页核验。

使用前置陈述：引理3.3 连续两次取边界为零。作者有明确编号回指。

证据：作者编号回指

Proof. This is obvious from lemma 3.3. Any element  $c$  of  $B_r(K)$  is written as  $c = \partial_{r+1}d$  for some  $d \in C_{r+1}(K)$ . Then we find  $\partial_r c = \partial_r(\partial_{r+1}d) = 0$ , that is,  $c \in Z_r(K)$ . This implies  $Z_r(K) \supset B_r(K)$ .

后文使用：定义3.5：同调群及作者的良定义说明。

定义3.5 同调群

对 n 维单纯复形，以循环群模边界群定义第 r 同调群。

条件与约定：本章主要采用整数系数；$0\le r\le n$。不能与实系数同调空间按定义完全等同。

符号：$H_r(K)=Z_r(K)/B_r(K)$

逐字作者来源与版本

Definition 3.5. Let  $K$  be an  $n$ -dimensional simplicial complex. The  $r$ th homology group  $H_{r}(K)$ ,  $0 \leq r \leq n$ , associated with  $K$  is defined by

$$
H _ {r} (K) \equiv Z _ {r} (K) / B _ {r} (K). \tag {3.28}
$$

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书库概念：同调群。 阅读背景：读到定义3.5：为什么能取这个商群（编辑导航，不计作者依赖）。

定义3.5：同调群及作者的良定义说明

以循环群模边界群定义同调群；作者随后用边界群包含于循环群说明商群良定义，并解释同调类。

条件与约定：本章整数系数单纯链；链群为自由阿贝尔群。该路径以群、子群和有向单形知识为已具备背景。 可另行采用实数或模二系数；不将不同系数的定义自动等同。

符号：$H_r(K)=Z_r(K)/B_r(K)$

逐字作者来源与版本

Definition 3.5. Let  $K$  be an  $n$ -dimensional simplicial complex. The  $r$ th homology group  $H_{r}(K)$ ,  $0 \leq r \leq n$ , associated with  $K$  is defined by

$$
H _ {r} (K) \equiv Z _ {r} (K) / B _ {r} (K). \tag {3.28}
$$

[Remarks: If necessary, we define  $H_r(K) = 0$  for  $r > n$  or  $r < 0$ . If we want to stress that the group structure is defined with integer coefficients, we

write  $H_{r}(K;\mathbb{Z})$ . We may also define the homology groups with  $\mathbb{R}$ -coefficients,  $H_{r}(K;\mathbb{R})$  or those with  $\mathbb{Z}_2$ -coefficients,  $H_{r}(K;\mathbb{Z}_{2})$ .]

Since  $B_r(K)$  is a subgroup of  $Z_r(K)$ ,  $H_r(K)$  is well defined. The group  $H_r(K)$  is the set of equivalence classes of  $r$ -cycles,

$$
H _ {r} (K) \equiv \{[ z ] | z \in Z _ {r} (K) \} \tag {3.29}
$$

where each equivalence class  $[z]$  is called a homology class. Two  $r$ -cycles  $z$  and  $z'$  are in the same equivalence class if and only if  $z - z' \in B_r(K)$ , in which case  $z$  is said to be homologous to  $z'$  and denoted by  $z \sim z'$  or  $[z] = [z']$ . Geometrically  $z - z'$  is a boundary of some space. By definition, any boundary  $b \in B_r(K)$  is homologous to 0 since  $b - 0 \in B_r(K)$ . We accept the following theorem without proof.

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使用前置陈述：定理3.3 边界群包含于循环群。作者的 Since 承接使用了定理3.3的包含关系，保证分母是分子中的子群；阿贝尔条件另由链群定义提供。

证据：作者承接

Since  $B_r(K)$  is a subgroup of  $Z_r(K)$ ,  $H_r(K)$  is well defined.

使用前置陈述：§3.1.1 商群 (3.2)–(3.3)。这里实际使用商群构造；本章阿贝尔条件保证无需另加一般群的正规性前提。此边是来源构造使用候选，不是编号回指。

证据：构造使用（待独立复核）

$$
H _ {r} (K) \equiv Z _ {r} (K) / B _ {r} (K). \tag {3.28}
$$

使用前置陈述：定义3.2：整数系数链群。链群的阿贝尔结构经子群继承，说明本处的商群条件；这不是把一般群的子群都当成正规子群。

证据：构造使用（待独立复核）

Since  $B_r(K)$  is a subgroup of  $Z_r(K)$ ,  $H_r(K)$  is well defined.

使用前置陈述：定义3.3：循环群是边界同态的核。商群分子 Zr 必须按作者的核定义理解，而非把所有链都当作循环。

证据：构造使用（待独立复核）

$$
H _ {r} (K) \equiv Z _ {r} (K) / B _ {r} (K). \tag {3.28}
$$

使用前置陈述：定义3.4：边界群是前一边界同态的像。商群分母 Br 必须按作者的像定义理解，商掉的是边界循环。

证据：构造使用（待独立复核）

$$
H _ {r} (K) \equiv Z _ {r} (K) / B _ {r} (K). \tag {3.28}
$$

使用前置陈述：循环群和边界群的子群资格。先确认 Zr、Br 的群资格，再结合定理3.3的包含关系组成商群。该前置结论单独不证明 Br 包含于 Zr。

证据：构造使用（待独立复核）

Since  $B_r(K)$  is a subgroup of  $Z_r(K)$ ,  $H_r(K)$  is well defined.

定义3.6 贝蒂数

贝蒂数取实系数同调群的维数，也等于整数同调自由部分的秩。

条件与约定：不得把整数同调的挠部分计入维数。

符号：$b_r(K)=\dim H_r(K;\mathbb{R})$

逐字作者来源与版本

Definition 3.6. Let  $K$  be a simplicial complex. The  $r$ th Betti number  $b_{r}(K)$  is defined by

$$
b _ {r} (K) \equiv \dim H _ {r} (K; \mathbb {R}). \tag {3.56}
$$

In other words,  $b_{r}(K)$  is the rank of the free Abelian part of  $H_{r}(K; \mathbb{Z})$ .

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定理3.7 欧拉–庞加莱关系

单形数的交错和等于贝蒂数的交错和；第一等式定义一般多面体的欧拉示性数。

条件与约定：有限 n 维单纯复形；$I_r$ 是 r 单形数，$b_r$ 为实系数同调维数。

符号：$\chi (K)$；$I_r$；$b_r(K)$

逐字作者来源与版本

Theorem 3.7. (The Euler-Poincaré theorem) Let  $K$  be an  $n$ -dimensional simplicial complex and let  $I_r$  be the number of  $r$ -simplexes in  $K$ . Then

$$
\chi (K) \equiv \sum_ {r = 0} ^ {n} (- 1) ^ {r} I _ {r} = \sum_ {r = 0} ^ {n} (- 1) ^ {r} b _ {r} (K). \tag {3.57}
$$

[Remark: The first equality defines the Euler characteristic of a general polyhedron  $|K|$ . Note that this is the generalization of the Euler characteristic defined for surfaces in section 2.4.]

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定理3.7证明中的维数分解

作者对实系数边界映射应用此前定理2.1。

条件与约定：该前置定理尚未绑定，本批不能用别的书替代。

逐字作者来源与版本

Proof. Consider the boundary homomorphism,

$$
\partial_ {r}: C _ {r} (K; \mathbb {R}) \to C _ {r - 1} (K; \mathbb {R})
$$

where  $C_{-1}(K; \mathbb{R})$  is defined to be  $\{0\}$ . Since both  $C_{r - 1}(K; \mathbb{R})$  and  $C_r(K;\mathbb{R})$  are vector spaces, theorem 2.1 can be applied to yield

$$
\begin{array}{l} I _ {r} = \dim C _ {r} (K; \mathbb {R}) = \dim (\ker \partial_ {r}) + \dim (\operatorname {i m} \partial_ {r}) \\ = \dim Z _ {r} (K; \mathbb {R}) + \dim B _ {r - 1} (K; \mathbb {R}) \\ \end{array}
$$

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