Homology groups (simplicial; simple definition)

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1 Introduction

We present a simplified definition of homology groups accessible to non-specialists in topology. Simple properties can be proved using this definition. For proving more advanced properties one may need more abstract reformulation, or more general definition. E.g. for Poincar\'e duality one needs a reformulation via a chain complex ($\S$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\S$4), and for topological invariance one needs singular homology.

The exposition follows [Skopenkov2015b, $\S$$\S$6, $\S$$\S$10]. See also expository papers [Dzhenzher&Miroshnikov&Nikitenko&Skopenkov2024], [Alkin&Dzhenzher&Nikitenko&Skopenkov&Voropaev2023].

Let $X$$X$ be a simplicial complex (or a cell complex in the sense of [Rourke&Sanderson1972]).

2 Definition for modulo 2 coefficients

A (simplicial, or cellular) $k$$k$-cycle modulo 2 in $X$$X$ is a set $x$$x$ of $k$$k$-faces such that every $(k-1)$$(k-1)$-face is contained in an even number of faces from $x$$x$. Consider the sum (modulo 2) operation on $k$$k$-cycle modulo 2 in $X$$X$.

If $\dim X=k$$\dim X=k$, then the homology group $H_k(X)$$H_k(X)$ modulo 2 is the group of homology classes of $k$$k$-cycles in $X$$X$.

In a general complex $X$$X$ two $k$$k$-cycles modulo 2 are homologous (modulo 2) if their sum (=difference) is the sum of boundaries of some $(k+1)$$(k+1)$-faces. The homology group $H_k(X)=H_k(X;\Z_2)$$H_k(X)=H_k(X;\Z_2)$ with $\Zz_2$$\Zz_2$-coefficients is the group of homology classes of $k$$k$-cycles in $X$$X$.

For different subdivisions $X$$X$ of a fixed complex $N$$N$ the groups $H_k(X)$$H_k(X)$ are isomorphic and are denoted by $H_k(N)$$H_k(N)$.

3 Definition for integer coefficients

Let $X$$X$ be a graph with oriented edges. An assignment of integers to oriented edges of $X$$X$ is a (simplicial) integer 1-cycle if for every vertex the sum of integers assigned to incoming edges equals the sum of integers assigned to outcoming edges (Kirchhof rule).

For $k>0$$k>0$, an orientation of a $k$$k$-simplex is an ordering of its vertices up to an even permutation. An orientation of a $0$$0$-simplex (i.e. of a vertex) is assignment of $+1$$+1$ or $-1$$-1$ to this $0$$0$-simplex. Alternatively, an orientation of a $k$$k$-simplex is a basis in a linear span of this simplex, up to orientation-preserving (in the sense of linear algebra) linear transformation. An oriented simplex is a simplex with some orientation. For an oriented simplex $\alpha$$\alpha$ denoted by $-\alpha$$-\alpha$ the same simplex with the opposite orientation.

For $k>0$$k>0$ let $\sigma=(\sigma_0,\ldots,\sigma_{k+1})$$\sigma=(\sigma_0,\ldots,\sigma_{k+1})$ be an oriented $k$$k$-simplex on vertices $\sigma_0,\ldots,\sigma_{k+1}$$\sigma_0,\ldots,\sigma_{k+1}$. For any $j\in\{0,\ldots,k+1\}$$j\in\{0,\ldots,k+1\}$ denote by $\widehat{\sigma}_j$$\widehat{\sigma}_j$ the oriented $(k-1)$$(k-1)$-face obtained by deleting $\sigma_j$$\sigma_j$ from $(\sigma_0,\ldots,\sigma_{k+1})$$(\sigma_0,\ldots,\sigma_{k+1})$. The oriented $k$$k$-simplex $\sigma$$\sigma$ comes in the $(k-1)$$(k-1)$-face $(-1)^j\widehat{\sigma}_j$$(-1)^j\widehat{\sigma}_j$, and comes out of the $(k-1)$$(k-1)$-face $(-1)^{j-1}\widehat{\sigma}_j$$(-1)^{j-1}\widehat{\sigma}_j$.

E.g.

• for $k=1$$k=1$ the oriented edge $AB=(A,B)$$AB=(A,B)$ comes in $+B,-A$$+B,-A$, and comes out of $+A,-B$$+A,-B$;

• for $k=2$$k=2$ the oriented 2-face $ABC=(A,B,C)$$ABC=(A,B,C)$ comes in $AB,BC,CA$$AB,BC,CA$, and comes out of $BA,CB,AC$$BA,CB,AC$.

This disagrees with other sign agreements, see e.g. [Skopenkov2018i, $\S$$\S$2.4.2], but both sign agreements work.

Coming in / out of $(k-1)$$(k-1)$-face depends on the orientation of the $(k-1)$$(k-1)$-face, but the properties described below do not depend of this orientation.

The (oriented) boundary of $\sigma$$\sigma$ is $\partial\sigma := \sum\limits_{j=0}^k (-1)^j\widehat{\sigma}_j$$\partial\sigma := \sum\limits_{j=0}^k (-1)^j\widehat{\sigma}_j$.

In a more standard terminology (not used below), for an oriented $(k-1)$$(k-1)$-face $\alpha$$\alpha$ define the incidence coefficient

$\displaystyle [\sigma:\alpha] := \begin{cases} \pm1 & \alpha=\pm(-1)^j\widehat{\sigma}_j \text{ for some }j \\ 0 & \text{otherwise (i.e. if $\alpha\not\subset\sigma$)} \end{cases}.$ $$\displaystyle [\sigma:\alpha] := \begin{cases} \pm1 & \alpha=\pm(-1)^j\widehat{\sigma}_j \text{ for some }j \\ 0 & \text{otherwise (i.e. if $\alpha\not\subset\sigma$)} \end{cases}.$$

Then $\partial\sigma = \sum\limits_{\alpha} [\sigma:\alpha]\alpha$$\partial\sigma = \sum\limits_{\alpha} [\sigma:\alpha]\alpha$.

Let $X$$X$ be a simplicial $k$$k$-complex whose $k$$k$-faces are oriented. An assignment of integers to oriented $k$$k$-faces of $X$$X$ is a (simplicial) integer $k$$k$-cycle in $X$$X$ if for every oriented $(k-1)$$(k-1)$-face the sum of integers assigned to incoming oriented $k$$k$-faces equals the sum of integers assigned to outcoming oriented $k$$k$-faces. (This is equivalent to the boundary of this assignment being zero, where the boundary is the homomorphism from assignments to integers defined as above on the basis.) E.g. the boundary of an oriented $(k+1)$$(k+1)$-face is an integer $k$$k$-cycle.

Consider the componentwise sum operation on integer $k$$k$-cycles in $X$$X$.

If $\dim X=k$$\dim X=k$, then the integer homology group $H_k(X;\Z)$$H_k(X;\Z)$ is the group of integer $k$$k$-cycles in $X$$X$.

In a general complex $X$$X$ two integer $k$$k$-cycles are homologous if their difference is a linear combination with integer coefficients of boundaries of some $(k+1)$$(k+1)$-faces. The integer homology group $H_k(X;\Z)$$H_k(X;\Z)$ is the group of homology classes of integer $k$$k$-cycles in $X$$X$.

For a cell complex $X$$X$ integer $k$$k$-cycle, boundary, and integer homology group are defined analogously. (The alternative definition of the orientation is used.)

4 Alternative definition via chain complex

We present the definition for $\Z_2$$\Z_2$-coefficients. Denote by $C_k(X)=C_k(X;\Z_2)$$C_k(X)=C_k(X;\Z_2)$ the set (the $\Z_2$$\Z_2$-space) of arrangements of zeroes and units on the $s$$s$-dimensional cells of $T$$T$ (so $C_k(X)=\{0\}$$C_k(X)=\{0\}$ if there are no $k$$k$-dimensional cells in $X$$X$). Denote by $\partial=\partial_k \colon C_k(X)\to C_{k-1}(X)$$\partial=\partial_k \colon C_k(X)\to C_{k-1}(X)$ the extension over $C_k(X)$$C_k(X)$ of the map taking a $k$$k$-dimensional cell $\sigma$$\sigma$ of $X$$X$ to the boundary of $\sigma$$\sigma$. Denote

$\displaystyle Z_k(X)=Z_k(X;\Z_2) := \ker\partial_k\quad\text{and}\quad B_k(X)=B_k(X;\Z_2):=\mathrm{im}\partial_{k+1}.$ $$\displaystyle Z_k(X)=Z_k(X;\Z_2) := \ker\partial_k\quad\text{and}\quad B_k(X)=B_k(X;\Z_2):=\mathrm{im}\partial_{k+1}.$$

Then define $H_k(X)=H_k(X;\Z_2) := Z_k(X)/B_k(X)$$H_k(X)=H_k(X;\Z_2) := Z_k(X)/B_k(X)$.

5 References

• [Alkin&Dzhenzher&Nikitenko&Skopenkov&Voropaev2023] E. Alkin, S. Dzhenzher, O. Nikitenko, A. Skopenkov, A. Voropaev, Cycles in graphs and in hypergraphs: results and problems.
• [Dzhenzher&Miroshnikov&Nikitenko&Skopenkov2024] S. Dzhenzher, A. Miroshnikov, O. Nikitenko, A. Skopenkov, Cycles in graphs and in hypergraphs.
• [Rourke&Sanderson1972] C. Rourke and B. Sanderson, Introduction to piecewise-linear topology., Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 69. Berlin-Heidelberg-New York: Springer-Verlag. VIII, 1972. MR0665919 (83g:57009) Zbl 0254.57010
• [Skopenkov2015b] A. Skopenkov, Algebraic Topology From Geometric Viewpoint (in Russian), MCCME, Moscow, 2015, 2020. Preprint of a part in English
• [Skopenkov2018i] A. Skopenkov, Invariants of graph drawings in the plane, Arnold Math. J., 6 (2020) 21-55. Full updated version: arXiv:1805.10237