Lefschetz Complexes as Finite Topological Spaces

TL;DR

This paper explores how a finite basis of a free chain complex can be viewed as a finite topological space, providing conditions under which their homologies are isomorphic, bridging algebraic and topological perspectives.

Contribution

It introduces a sufficient condition linking the homology of a chain complex with the homology of its associated finite topological space.

Findings

Homology of the space matches the chain complex under certain conditions

Provides a new perspective on finite topological spaces and chain complexes

Bridges algebraic and topological methods in homology theory

Abstract

We consider a fixed basis of a finitely generated free chain complex as a finite topological space and we present a sufficient condition for the singular homology of this space to be isomorphic with the homology of the chain complex.

Equations

$${\mathrm{opn}}_{\text{T}}A\text{mathchar}58=\bigcap \{U\in \text{T}\mid A\subset U\}$$

$$\text{$\mathcal{T}$}_{\leq}\mathrel{\mathop{\mathchar 58\relax}}=\mbox{$\{\,A\subset X\mid\text{ $A$ is an upper set with respect to $\leq$}\,\}$}$$

$$\leq_{\text{$\mathcal{T}$}}\mathrel{\mathop{\mathchar 58\relax}}=\mbox{$\{\,(x,y)\in X^{2}\mid x\in\operatorname{cl}_{\text{$\mathcal{T}$}}y\,\}$}.$$

$$x^{\ge }={\mathrm{opn}}_{{\text{T}}_{\le }}x.$$

$$x^{{\ge }_{\text{T}}}={\mathrm{opn}}_{\text{T}}x.$$

$$\alpha \text{mathchar}58\text{textsc}PreOrd(X)\ni \le$$

$$\beta \text{mathchar}58\text{textsc}Top(X)\ni \text{T}$$

$$\sum _{y\in X}\kappa (x,y)\kappa (y,z)=0.$$

$$j(c)=\{\begin{array}{ll}0,& c\in C^{\kappa }(X^{\prime }),\\ c,& \text{otherwise}.\end{array}$$

$$\begin{diagram}$$

$$c_x\text{mathchar}58{\Delta }_0=\{1\}\ni 1\mapsto x\in X_0$$

$$\forall c\in C^{\kappa }(X)|\phi (c)|\subset \mathrm{cl}|c|.$$

$$H_n^{\kappa }(\mathrm{cl}x)=\{\begin{array}{ll}R& n=0,\\ 0& \text{otherwise.}\end{array}$$

$$\begin{diagram}.$$

$$H_n^{\kappa }(X,X^{\prime })\cong H_n^{\kappa }(X\setminus X^{\prime })=H_n^{\kappa }(a)=H_n^{\kappa }(\mathrm{cl}a\setminus \mathrm{mo}a)\cong H_n^{\kappa }(\mathrm{cl}a,\mathrm{mo}a).$$

$$\begin{diagram}.$$

$$H_n^{\kappa }(\mathrm{cl}e)=H_n^{\kappa }(X)=\{\begin{array}{ll}{R}^3& n=0,\\ 0& \text{otherwise.}\end{array},$$

$$H_n(X)=\{\begin{array}{ll}R& n=0,\\ 0& \text{otherwise.}\end{array}$$

Full text

This research is partially supported by the Polish National Science Center under Maestro Grant No. 2014/14/A/ST1/00453.

1. Introduction

The combinatorial Morse theory, originally developed for finite CW complexes by R. Forman [4] in 1998, was generalized to the purely algebraic setting of finitely generated free chain complexes with a distinguished basis by Kozlov [11] in 2005, and independently by Skjöldberg [19] in 2006 and Jöllenbeck and Welker [8] in 2009. A finitely generated free chain complex with a distinguished basis was already studied by Lefschetz in 1942. However, in the original Lefschetz definition [12, Chpt. III, Sec. 1, Def. 1.1] the elements of the basis are the primary objects of interest and the algebraic structure of the associated chain complex is given on top of them (see Section 2.3 for a precise definition).

Thus, the Lefschetz complex consists of a finite collection of cells $X$ graded by dimension and the incidence coefficient $\kappa(x,y)$ encoding the incidence relation between cells $x,y\in X$. This way of viewing a free chain complex with a distinguished basis is convenient in the algorithmic context, because a computer may store and process only finite sets. In particular, in recent years, Lefschetz complexes became an object of interest as a convenient tool in computational dynamics and topology [6, 7, 17]. As examples of Lefschetz complexes let us mention: the family $K$ of all simplices of a simplicial complex [9, Definition 11.8], all elementary cubes in a cubical set [9, Definition 2.9] or, more generally, cells of a cellular complex (finite CW complex, see [13, Section IX.3]). A non-zero value of $\kappa(x,y)$ indicates that the cell $y$ is in the boundary of the cell $x$ and the dimension of $y$ is one less than the dimension of $x$. The cell $y$ is then called a facet of $x$. The linear combination of facets of a cell $x$ with incidence coefficients as coefficients is the boundary of $x$. Under suitable assumptions this defines the associated free chain complex with basis $X$ and the homology of $X$ which we will refer to as the Lefschetz homology of $X$.

The relation $y\prec_{\kappa}x$ defined by $\kappa(x,y)\neq 0$ extends to a partial order $\leq_{\kappa}$ on $X$. Hence, by Alexandroff Theorem [1], a Lefschetz complex is also a finite topological space. Therefore, also its singular homology is well defined. This leads to a natural question under what conditions the Lefschetz homology is isomorphic to the singular homology. In this paper we give a partial answer to this question by proving the following theorem (see Theorem 3.2 for the precise formulation).

Theorem**.**

Let $(X,\kappa)$ be an augmentable Lefschetz complex such that the Lefschetz homology of the closure of every point in $X$ coincides with the singular homology of the one-point space. Then the Lefschetz homology and the singular homology of $X$ are isomorphic.

The organization of the paper is as follows. In Section 2 we recall some preliminary notions and theorems about chain complexes, finite topological spaces and Lefschetz complexes. In Section 3 we state and prove the main result of this paper.

2. Preliminaries

In this section we recall definitions and results needed in the sequel and we fix the notation.

2.1. Chain complexes.

Recall that given a graded free module $C=(C_{n})_{n\in\mathbb{Z}}$ (i.e. $C_{n}$ is a free module for each $n\in\mathbb{Z}$) and a sequence $\partial=(\partial_{n}\mathrel{\mathop{\mathchar 58\relax}}C_{n}\rightarrow C_{n-1})_{n\in\mathbb{Z}}$ such that for each $n\in\mathbb{Z}$ we have $\partial_{n}\partial_{n-1}=0$, the pair $(C,\partial)$ is called a chain complex and $\partial$ is called the associated boundary map.

Given a chain complex $(C,\partial)$ and an $n\in\mathbb{Z}$ we write $B_{n}(C)\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{im}\partial_{n+1}$ for the $n$-boundaries and $Z_{n}(C)\mathrel{\mathop{\mathchar 58\relax}}=\ker\partial_{n}$ for the $n$-cycles. The fact that $B_{n}(C)$ is a subgroup of $Z_{n}(C)$ lets us define the $n$-th homology group as $H_{n}(C)\mathrel{\mathop{\mathchar 58\relax}}=Z_{n}(C)/B_{n}(C)$.

Let $(C,\partial)$ and $(C^{\prime},\partial^{\prime})$ be chain complexes. A sequence $(\varphi_{n}\mathrel{\mathop{\mathchar 58\relax}}C_{n}\rightarrow C^{\prime}_{n})_{n\in\mathbb{Z}}$ of maps is called a chain map if $\varphi_{n-1}\partial_{n}=\partial_{n}\varphi_{n}$ for each n\in\mbox{\mathbb{Z}}.

In the sequel we drop the subscripts in boundary maps and chain maps whenever they are clear from the context.

The following excision theorem for simplicial complexes will be used in the sequel.

Theorem 2.1**.**

[3, Ch. V, Sec. 2, Thm. 4]** If \text{\mathcal{K}},\text{\mathcal{K}}_{1},\text{\mathcal{K}}_{2} are simplicial complexes such that \text{\mathcal{K}}=\text{\mathcal{K}}_{1}\cup\text{\mathcal{K}}_{2}, then H(|\text{\mathcal{K}}_{2}|,|\text{\mathcal{K}}_{1}\cap\text{\mathcal{K}}_{2}|)=H(|\text{\mathcal{K}}|,|\text{\mathcal{K}}_{1}|).

2.2. Finite topological spaces.

Recall that a finite topological space is a pair (X,\text{\mathcal{T}}) such that $X$ is a finite set and $\mathcal{T}$ is a topology on $X$.

Note that, unlike for the general topological spaces, if (X,\text{\mathcal{T}}) is a finite topological space and $A\subset X$, then

[TABLE]

is open.

To keep the notation simple we write \operatorname{cl}_{\text{\mathcal{T}}}x\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{cl}_{\text{\mathcal{T}}}\{x\},\operatorname{opn}_{\text{\mathcal{T}}}x\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{opn}_{\text{\mathcal{T}}}\{x\} for any $x\in X$. Also, whenever topology $\mathcal{T}$ is clear from the context, we drop the subscript $\mathcal{T}$ in \operatorname{cl}_{\text{\mathcal{T}}} and \operatorname{opn}_{\text{\mathcal{T}}}.

Consider a finite set $X$. Denote by $\textsc{PreOrd}(X)$ the family of all preorders on $X$, by $\textsc{PartOrd}(X)$ the family of all partial orders on $X$, by $\textsc{Top}(X)$ the family of all topologies on $X$ and by $\textsc{Top}_{0}(X)$ the subfamily of $T_{0}$ topologies on $X$. For a preorder $\leq$ in $\textsc{PreOrd}(X)$ and a subset $A\subset X$ the set $A^{\leq}\mathrel{\mathop{\mathchar 58\relax}}=\{x\in X\mid\exists a\in A\;x\leq a\}$ is called the lower set of $A$ and $A^{\geq}\mathrel{\mathop{\mathchar 58\relax}}=\{x\in X\mid\exists a\in A\;a\leq x\}$ is called the upper set of $A$. For convenience we set $x^{\leq}\mathrel{\mathop{\mathchar 58\relax}}=\{x\}^{\leq}$ and $x^{\geq}\mathrel{\mathop{\mathchar 58\relax}}=\{x\}^{\geq}$. We say that $A$ is an upper (respectively lower) set if $A=A^{\leq}$ (respectively $A=A^{\geq}$).

For a preorder $\leq$ in $\textsc{PreOrd}(X)$ set

[TABLE]

and for a topology \text{\mathcal{T}}\in\textsc{Top}(X) set

[TABLE]

Finite topological spaces can be characterized in terms of preorders as in the following theorem which goes back to P.S. Alexandroff [1].

Theorem 2.2**.**

Let $X$ be a finite set. We have the following properties.

• (i)

For every preorder $\leq$ in $\textsc{PreOrd}(X)$ the family \text{\mathcal{T}}_{\leq} is a topology on $X$. Moreover, for every $x\in X$

[TABLE]

• (ii)

For every topology \text{\mathcal{T}}\in\textsc{Top}(X) the relation \geq_{\text{\mathcal{T}}} defined by $x\in\operatorname{cl}y$ is a preorder on $X$. Moreover, for any $x\in X$

[TABLE]

• (iii)

The maps

[TABLE]

are mutually inverse bijections under which the partial orders on $X$ correspond to $T_{0}$ topologies on $X$.

Theorem 2.3**.**

[2, Prop. 1.2.1]** Assume (X,\text{\mathcal{T}}),(Y,\text{\mathcal{S}}) are finite topological spaces. The map f\mathrel{\mathop{\mathchar 58\relax}}(X,\text{\mathcal{T}})\rightarrow(Y,\text{\mathcal{S}}) is continuous if and only if the map f\mathrel{\mathop{\mathchar 58\relax}}\left(X,\leq_{\text{\mathcal{T}}}\right)\rightarrow(Y,\leq_{\text{\mathcal{S}}}) is order preserving.

Theorem 2.4**.**

[2*, Rem. 1.2.8]**

Let (X,\text{\mathcal{T}}) be a finite $T_{0}$ topological space. Then $\operatorname{cl}x$ is contractible for any $x\in X$.*

Now assume (X,\text{\mathcal{T}}) is a finite $T_{0}$ space and denote by \text{\mathcal{K}}(X) the abstract simplicial complex of subsets of $X$ linearly ordered by \leq_{\text{\mathcal{T}}} and its geometric realization by \mathinner{\!\left\lvert\text{\mathcal{K}}(X)\right\rvert}. The McCord map is the map $\mu\mathrel{\mathop{\mathchar 58\relax}}\mathinner{\!\left\lvert\mathcal{K}(X)\right\rvert}\rightarrow X$ defined for a point p=t_{1}x_{1}+...+t_{k}x_{k}\in\mathinner{\!\left\lvert\text{\mathcal{K}}(X)\right\rvert} by $\mu(p)\mathrel{\mathop{\mathchar 58\relax}}=x_{1}$ where x_{1}\leq_{\text{\mathcal{T}}}x_{2}\leq_{\text{\mathcal{T}}}...\leq_{\text{\mathcal{T}}}x_{k}.

Theorem 2.5**.**

(McCord, **[14]) For every finite $T_{0}$ topological space $X$ the McCord map $\mu$ is a weak homotopy equivalence.

Corollary 2.6**.**

For every finite $T_{0}$ space $X$ the McCord map $\mu$ induces an isomorphism \mu_{*}\mathrel{\mathop{\mathchar 58\relax}}H(|\text{\mathcal{K}}(X)|)\rightarrow H(X).

Using elementary results from homological algebra we obtain the following relative version of McCord Theorem.

Corollary 2.7**.**

Let $X$ be a finite $T_{0}$ space and let $A\subset X$ be its subspace. Then the McCord map $\mu$ induces an isomorphism from $H(X,A)$ to $H(|\mathcal{K}(X)|,|\mathcal{K}(A)|)$.

2.3. Lefschetz complexes.

Let $R$ denote a ring with unity. We say that $(X,\kappa)$ is a Lefschetz complex (see [12, Chpt. III, Sec. 1, Def. 1.1] and compare also with definition of an S-complex in [16, Section 2]). If $X=(X_{q})_{q\in{\scriptsize\mathbb{Z}^{+}}}$ is a finite set with gradation, $\kappa\mathrel{\mathop{\mathchar 58\relax}}X\times X\to R$ is a map such that $\kappa(x,y)\neq 0$ implies $x\in X_{q},\,y\in X_{q-1}$ and for any $x,z\in X$ we have

[TABLE]

We refer to the elements of $X$ as cells and to $\kappa(x,y)$ as the incidence coefficient of $x,y$. If $x\in X_{q}$, we say that $x$ has dimension $q$.

Let $(X,\kappa)$ be a fixed Lefschetz complex. Denote by $C^{\kappa}(X)$ the graded free module over $R$ spanned by $X$. We refer to the elements of $C^{\kappa}(X)$ as Lefschetz chains. Define $\partial^{\kappa}\mathrel{\mathop{\mathchar 58\relax}}C^{\kappa}(X)\rightarrow C^{\kappa}(X)$ on generators $x\in X$ by $\partial^{\kappa}(x)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{y\in X}\kappa(x,y)y$. By (3) we easily see that $(C^{\kappa}(X),\partial^{\kappa})$ is a chain complex. We denote its $n$-th homology group by $H_{n}^{\kappa}(X)$ and call it $n$-th Lefschetz homology group of $X$.

Note that the concept of Lefschetz complex is complementary to that of chain complex: it shifts focus to the basis of the complex. Indeed, if $(C,\partial)$ is a finitely generated cell complex and $(e_{i})_{i=1}^{n}$ is its fixed basis, then by taking $X=\{e_{i}\mid i=1,\ldots,n\}$ and setting $\kappa(e_{i},e_{j})$ as the coefficient of $e_{j}$ in the decomposition of $\partial e_{i}$ on the basis, we obtain Lefschetz complex $X$ such that $(C^{\kappa}(X),\partial^{\kappa})=(C,\partial)$.

If for $x,y\in X$ we have $\kappa(x,y)\neq 0$, we say that $y$ is a facet of $x$ and we write $y\prec x$. We denote by $\leq$ the transitive and reflexive closure of $\prec$, that is the smallest with respect to inclusion relation that contains $\prec$ and is both transitive and reflexive. If $y\leq x$ we say that $y$ is a face of $x$.

By Theorem 2.2 we have the $T_{0}$-topology \text{\mathcal{T}}_{\leq} associated with $\left(X,\leq\right)$. We call it the Lefschetz topology of $X$. By $C(X)$ we denote the chain complex of singular chains on (X,\text{\mathcal{T}}_{\leq}). The chain complexes of singular chains $C(X)$ should not be mistaken with the chain complex of Lefschetz chains, denoted by $C^{\kappa}(X)$.

Now, we recall a few definitions and theorems which will be used in the sequel.

Given $A\subset X$ we define the mouth of $A$ as the set $\operatorname{mo}A\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{cl}A\setminus A$. We say that $A$ is locally closed if $\operatorname{mo}A$ is closed. Note that in [17] a locally closed set is called proper. Observe that, in particular, closed and open sets are locally closed.

Theorem 2.8**.**

[16, Theorem 3.1]** Let $(X,\kappa)$ be a Lefschetz complex and let $A\subset X$ be locally closed in the Lefschetz topology of $X$. Then $(A,\kappa\restriction_{A\times A})$ is a Lefschetz complex.

Theorem 2.9**.**

[16, Theorem 3.4]** Assume $X$ is a Lefschetz complex and $X^{\prime}\subset X$ is closed. Let $i\mathrel{\mathop{\mathchar 58\relax}}C^{\kappa}(X^{\prime})\rightarrow C^{\kappa}(X)$ denote inclusion and $j\mathrel{\mathop{\mathchar 58\relax}}C^{\kappa}(X)\rightarrow C^{\kappa}(X\setminus X^{\prime})$ be the projection defined by

[TABLE]

Then the following sequence

[TABLE]

is exact and it extends to a long exact sequence.

Theorem 2.10**.**

[16, Theorem 3.5]** Let $(X,\kappa)$ be a Lefschetz complex and $X^{\prime}\subset X$ be closed in the Lefschetz topology of $X$. Then the homology of $H^{\kappa}(X,X^{\prime})$ the quotient chain complex $(C^{\kappa}(X)/C^{\kappa}(X^{\prime}),\partial^{\kappa})$ is isomorphic to $H^{\kappa}(X\setminus X^{\prime})$.

3. Main result

Let $(X,\kappa)$ be a Lefschetz complex over a ring $R$. We say that $(X,\kappa)$ is augmentable, if the linear map $\epsilon^{\kappa}\mathrel{\mathop{\mathchar 58\relax}}C_{0}^{\kappa}\rightarrow R$ defined on the basis elements $x\in X_{0}$ by $\epsilon^{\kappa}(x)=1$ satisfies $\epsilon^{\kappa}\partial_{1}^{\kappa}=0$. The following lemma may be deduced from the Acyclic Carrier Theorem [18, Thm. 13.4]. We include a direct proof for convenience.

Lemma 3.1**.**

Assume that $(X,\kappa)$ is an augmentable Lefschetz complex. Then there exists a chain map $\varphi\mathrel{\mathop{\mathchar 58\relax}}C^{\kappa}(X)\rightarrow C(X)$ from Lefschetz chains to singular chains such that $\varphi_{0}$ on the basis element $x\in X_{0}$ is the singular chain

[TABLE]

and

[TABLE]

**Proof: ** For $k<0$ we set $\varphi_{k}\mathrel{\mathop{\mathchar 58\relax}}=0$ and we define $\varphi_{0}$ on the basis element $x\in X_{0}$ as $c_{x}\in C_{0}(X)$ where $c_{x}$ is given by (4). Obviously $\partial_{0}\varphi_{0}=0=\varphi_{-1}\partial_{0}^{\kappa}$. Also (5) is trivially satisfied for $k=0$.

Thus, consider $k>0$ and assume that $\varphi\mathrel{\mathop{\mathchar 58\relax}}C_{k-1}^{\kappa}(X)\rightarrow C_{k-1}(X)$ satisfying $\partial_{k-1}\varphi_{k-1}=\varphi_{k-2}\partial_{k-1}^{\kappa}$ and (5) is already defined. For $x\in X_{k}$ set $u_{x}\mathrel{\mathop{\mathchar 58\relax}}=\varphi_{k-1}\partial_{k}^{\kappa}x\in C_{k-1}(X)$. Then $\partial_{k-1}u_{x}=\partial_{k-1}\varphi_{k-1}\partial_{k}^{\kappa}x=\varphi_{k-2}\partial_{k-1}^{\kappa}\partial_{k}^{\kappa}x=0$. Hence, $u_{x}$ is a singular $(k-1)$-cycle. Since $\varphi_{k-1}$ satisfies (5), we also have $|u_{x}|=|\varphi_{k-1}\partial_{k}^{\kappa}x|\subset\operatorname{cl}|\partial_{k}^{\kappa}x|\subset\operatorname{cl}\operatorname{cl}x=\operatorname{cl}x$. It follows that $u_{x}\in Z_{k-1}(\operatorname{cl}x)$. If $k>1$, then $u_{x}\in B_{k-1}(\operatorname{cl}x)$, because by Theorem 2.4 the set $\operatorname{cl}x$ is contractible, which means that $H_{l}(\operatorname{cl}x)=0$ for $l>0$. Hence, consider the case $k=1$. Let $\epsilon\mathrel{\mathop{\mathchar 58\relax}}C_{0}(X)\rightarrow R$ denote the augmentation for the singular chains. Then, for $x\in X_{0}$ we have $\epsilon\varphi_{0}x=\epsilon c_{x}=1=\epsilon^{\kappa}x$. It follows, that we have $\epsilon\varphi_{0}=\epsilon^{\kappa}$ and we get $\epsilon u_{x}=\epsilon\varphi_{0}\partial_{1}^{\kappa}x=\epsilon^{\kappa}\partial_{1}^{\kappa}x=0$ by the assumed augmentability of $(X,\kappa)$. Moreover, since $\operatorname{cl}x$ is contractible, we have $\ker\epsilon=\operatorname{im}\partial_{1}$. It means that $u_{x}\in B_{0}(\operatorname{cl}x)$.

Thus, we have proved that for every $k>0$ and $x\in X_{k}$ there exists a chain $s_{x}\in C_{k}(\operatorname{cl}x)$ such that $u_{x}=\partial s_{x}$. Set $\varphi_{k}(x)\mathrel{\mathop{\mathchar 58\relax}}=s_{x}$ for $x\in X_{k}$ and extend it linearly to $\varphi_{k}\mathrel{\mathop{\mathchar 58\relax}}C_{k}^{\kappa}(X)\rightarrow C_{k}(X)$. We have $|\varphi_{k}(x)|=|s_{x}|\subset\operatorname{cl}x$ and $\partial_{k}\varphi_{k}x=\partial_{k}s_{x}=u_{x}=\varphi_{k-1}\partial_{k}x$ for every $x\in X_{k}$. Now consider a Lefschetz chain $c=\sum_{i=1}^{n}\alpha_{i}x_{i}\in C_{k}^{\kappa}(X)$ where $n\in\mathbb{N},\alpha_{i}\in R,x_{i}\in X_{k}$ for $i=1,...,n$. We have $|\varphi_{k}(c)|\subset\bigcup_{i=1}^{n}|\varphi_{k}(x_{i})|\subset\bigcup_{i=1}^{n}\operatorname{cl}x_{i}=\operatorname{cl}\bigcup_{i=1}^{n}\{x_{i}\}=\operatorname{cl}|c|$. Therefore, $\varphi_{k}$ satisfies (5). We also have $\partial_{k}\varphi_{k}(c)=\sum_{i=1}^{n}\alpha_{i}\partial_{k}\varphi_{k}(x_{i})=\sum_{i=1}^{n}\alpha_{i}\varphi_{k-1}\partial_{k}(x_{i})=\varphi_{k-1}\partial_{k}\sum_{i=1}^{n}\alpha_{i}x_{i}=\varphi_{k-1}\partial_{k}c$. This concludes the proof. ∎

Now we are ready to state the main theorem of this paper.

Theorem 3.2**.**

Let $(X,\kappa)$ be an augmentable Lefschetz complex such that for every $x\in X$ we have

[TABLE]

Then $H^{\kappa}(X)\cong H(X)$.

**Proof: ** Let $\varphi\mathrel{\mathop{\mathchar 58\relax}}C^{\kappa}(X)\rightarrow C(X)$ be a chain map such as in Lemma 3.1. We will prove by induction with respect to $k\mathrel{\mathop{\mathchar 58\relax}}=\operatorname{card}X$ that $\varphi_{*}$ is an isomorphism. If $k=1$, then the only element in $X$ is of dimension zero, because otherwise (6) is not satisfied and the conclusion follows immediately. Assume now that $k>1$ and that thesis holds for the Lefschetz complexes of cardinality $k^{\prime}<k$. Let $a$ be a maximal element in $X$ with respect to $\leq$ and set $X^{\prime}\mathrel{\mathop{\mathchar 58\relax}}=X\setminus\{a\}$. Then $X^{\prime}$ is closed in $X$. Thus, by Theorem 2.8 we have a well defined Lefschetz complex $(X^{\prime},\kappa_{X^{\prime}\times X^{\prime}})$. It is augmentable, because its boundary operator is the restriction of the boundary operator on $X$, and it satisfies (6), because maximality of $a$ implies that for every $x\in X^{\prime}$ its closure is the same in $X$ and $X^{\prime}$. Therefore, $(X^{\prime},\kappa_{X^{\prime}\times X^{\prime}})$ satisfies the inductive assumption.

Observe that since 5 holds, $\varphi(C^{\kappa}(X^{\prime}))\subset C(X^{\prime})$ and consequently $\varphi$ induces a chain map $\varphi_{*}\mathrel{\mathop{\mathchar 58\relax}}H^{\kappa}(X,X^{\prime})\rightarrow H(X,X^{\prime})$. Thus, by Lemma 3.1 and Theorem 2.9 we have the following commutative diagram with exact rows

[TABLE]

By the induction assumption $\varphi_{*}\mathrel{\mathop{\mathchar 58\relax}}H_{n}^{\kappa}(X^{\prime})\rightarrow H_{n}(X^{\prime})$ is an isomorphism for every $n\in\mathbb{Z}$.

By Corollary 2.7 we have $H_{n}(X,X^{\prime})\cong H_{n}(|\mathcal{K}(X)|,|\mathcal{K}(X^{\prime})|)$. Observe that $\mathcal{K}(\operatorname{cl}a)$ is a subcomplex of $\mathcal{K}(X)$ such that $\mathcal{K}(X)=\mathcal{K}(X^{\prime})\cup\mathcal{K}(\operatorname{cl}a)$. Hence, $H_{n}(\mathcal{K}(X),\mathcal{K}(X^{\prime}))\cong H_{n}(\mathcal{K}(\operatorname{cl}a),\mathcal{K}(\operatorname{cl}a)\cap\mathcal{K}(X^{\prime}))=H_{n}(\mathcal{K}(\operatorname{cl}a),\mathcal{K}(\operatorname{mo}a))$ by Theorem 2.1. Thus, using Corollary 2.7 again we get $H_{n}(X,X^{\prime})\cong H_{n}(\operatorname{cl}a,\operatorname{mo}a)$. We also get from Theorem 2.10 that

[TABLE]

Now consider the diagram

[TABLE]

The homomorphism $\varphi_{*}\mathrel{\mathop{\mathchar 58\relax}}H_{n}^{\kappa}(\operatorname{mo}a)\rightarrow H_{n}(\operatorname{mo}a)$ is an isomorphisms for $n\in\mathbb{Z}$ by the induction assumption. For $n\in\mathbb{Z}\setminus\{1\}$ map $\varphi_{*}\mathrel{\mathop{\mathchar 58\relax}}H_{n}^{\kappa}(\operatorname{cl}x)=0\rightarrow H_{n}(\operatorname{cl}x)=0$ is obviously an isomorphism and for $n=0$ map $\varphi_{*}\mathrel{\mathop{\mathchar 58\relax}}H_{n}^{\kappa}(\operatorname{cl}x)=R\rightarrow H_{n}(\operatorname{cl}x)=R$ is an isomorphism straightforwardly from the definition of $\varphi$. By the Five Lemma (see [13, Lemma 7.1]) it follows that $\varphi_{*}\mathrel{\mathop{\mathchar 58\relax}}H_{n}^{\kappa}(\operatorname{cl}a,\operatorname{mo}a)\rightarrow H_{n}(\operatorname{cl}a,\operatorname{mo}a)$ is an isomorphism. Again by the Five Lemma we obtain from diagram LABEL:dg:main-thm-diagram that $\varphi_{*}\mathrel{\mathop{\mathchar 58\relax}}H_{n}^{\kappa}(X)\rightarrow H_{n}(X)$ is also an isomorphisms. This completes the proof. ∎

Note that a closed subcomplex of the augmentable Lefschetz complex is itself augmentable. Thus, we easily obtain the following corollary

Corollary 3.3**.**

Let $(X,\kappa)$ be an augmentable Lefschetz complex. Then $(X,\kappa)$ satisfies (6) if and only if for every closed subcomplex $X^{\prime}\subset X$ we have $H^{\kappa}(X^{\prime})\cong H(X^{\prime})$.

Corollary 3.3 does not resolve the problem whether inverse to Theorem 3.2 is true. This question remains open.

Now we provide some examples, which show that assumptions of Theorem 3.2 are essential. Here we take our ring to be R=\mbox{\mathbb{Z}}.

Example 3.4**.**

Consider a Lefschetz complex $(X,\kappa)$ with $X_{0}=\{a,b,c,d\}$, $X_{1}=\{e\}$ and incidence coefficients $\kappa(e,a)=1,\kappa(e,b)=1,\kappa(e,c)=-1,\kappa(e,d)=-1$ given by the matrix

[TABLE]

.

This Lefschetz complex and the associated simplicial complex is visualized in Figure 1. It is elementary to observe that $X$ is augmentable and

[TABLE]

However, by Theorem 2.4 we have

[TABLE]

Example 3.5**.**

Consider a Lefschetz complex $(X,\kappa)$ with $X_{0}=\{a,b\}$, $X_{1}=\{c,d\}$ and incidence coefficients given by the matrix

[TABLE]

This Lefschetz complex and the associated simplicial complex is visualized in Figure 1. One easily verifies that $X$ satisfies assumption (6), but it is not augmentable. This Lefschetz complex has no non-trivial cycles in dimension $1$. Indeed, assume that $z\mathrel{\mathop{\mathchar 58\relax}}=\gamma c+\delta d\in Z_{1}^{\kappa}(X)$ where \gamma,\delta\in\mbox{\mathbb{Z}}. It follows that $0=\partial z=(\gamma+\delta)a+(\delta-\gamma)b$. This is true only if $\gamma=\delta=0$. Thus, $z=0$ and consequently $H_{1}^{\kappa}(X)=0$. However, by Corollary (2.6) we easily obtain that $H_{1}(X)\cong H_{1}(\mathbb{S}^{1})\cong R$.