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{{Authors|Askopenkov}} {{Stub}} == Introduction == <wikitex>; 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 [[Intersection_form#Some_properties|Poincar\'e duality]] one needs a reformulation via a chain complex ($\S$\ref{s:alt}), and for topological invariance one needs ''singular homology''. The exposition follows \cite[$\S$6, $\S$10]{Skopenkov2015b}. See also expository papers \cite{Dzhenzher&Miroshnikov&Nikitenko&Skopenkov2024}, \cite{Alkin&Dzhenzher&Nikitenko&Skopenkov&Voropaev2023}. Let $X$ be a simplicial complex (or a cell complex in the sense of \cite{Rourke&Sanderson1972}). </wikitex> == Definition for modulo 2 coefficients== <wikitex>; A (simplicial, or cellular) '''$k$-cycle modulo 2''' in $X$ is a set $x$ of $k$-faces such that every $(k-1)$-face is contained in an even number of faces from $x$. Consider the sum (modulo 2) operation on $k$-cycle modulo 2 in $X$. If $\dim X=k$, then the ''homology group'' $H_k(X)$ ''modulo 2'' is the group of homology classes of $k$-cycles in $X$. In a general complex $X$ two $k$-cycles modulo 2 are ''homologous'' (modulo 2) if their sum (=difference) is the sum of boundaries of some $(k+1)$-faces. The '''homology group''' $H_k(X)=H_k(X;\Z_2)$ '''with $\Zz_2$-coefficients''' is the group of homology classes of $k$-cycles in $X$. For different subdivisions $X$ of a fixed complex $N$ the groups $H_k(X)$ are isomorphic and are denoted by $H_k(N)$. </wikitex> == Definition for integer coefficients== <wikitex>; Let $X$ be a graph with oriented edges. An assignment of integers to oriented edges of $X$ 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 $k>0$, an ''orientation'' of a $k$-simplex is an ordering of its vertices up to an even permutation. An ''orientation'' of a $0$-simplex (i.e. of a vertex) is assignment of $+1$ or $-1$ to this $0$-simplex. Alternatively, an ''orientation'' of a $k$-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 $\alpha$ denoted by $-\alpha$ the same simplex with the opposite orientation. For $k>0$ let $\sigma=(\sigma_0,\ldots,\sigma_{k+1})$ be an oriented $k$-simplex on vertices $\sigma_0,\ldots,\sigma_{k+1}$. For any $j\in\{0,\ldots,k+1\}$ denote by $\widehat{\sigma}_j$ the oriented $(k-1)$-face obtained by deleting $\sigma_j$ from $(\sigma_0,\ldots,\sigma_{k+1})$. The oriented $k$-simplex $\sigma$ ''comes in'' the $(k-1)$-face $(-1)^j\widehat{\sigma}_j$, and ''comes out of'' the $(k-1)$-face $(-1)^{j-1}\widehat{\sigma}_j$. E.g. * for $k=1$ the oriented edge $AB=(A,B)$ comes in $+B,-A$, and comes out of $+A,-B$; * for $k=2$ the oriented 2-face $ABC=(A,B,C)$ comes in $AB,BC,CA$, and comes out of $BA,CB,AC$. This disagrees with other sign agreements, see e.g. \cite[$\S$2.4.2]{Skopenkov2018i}, but both sign agreements work. Coming in / out of $(k-1)$-face depends on the orientation of the $(k-1)$-face, but the properties described below do not depend of this orientation. The (oriented) '''boundary''' of $\sigma$ is $\partial\sigma := \sum\limits_{j=0}^k (-1)^j\widehat{\sigma}_j$. In a more standard terminology (not used below), for an oriented $(k-1)$-face $\alpha$ define the ''incidence coefficient'' $$[\sigma:\alpha] := \begin{cases} \pm1 & \alpha=\pm(-1)^j\widehat{\sigma}_j \text{ for some }j \\ 0 & \text{otherwise (i.e. if $\alpha\not\subset\sigma$)} \end{cases}.$$ Then $\partial\sigma = \sum\limits_{\alpha} [\sigma:\alpha]\alpha$. Let $X$ be a simplicial $k$-complex whose $k$-faces are oriented. An assignment of integers to oriented $k$-faces of $X$ is a (simplicial) '''integer $k$-cycle''' in $X$ if for every oriented $(k-1)$-face the sum of integers assigned to incoming oriented $k$-faces equals the sum of integers assigned to outcoming oriented $k$-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 $(k+1)$-face is an integer $k$-cycle. Consider the componentwise sum operation on integer $k$-cycles in $X$. If $\dim X=k$, then the ''integer homology group'' $H_k(X;\Z)$ is the group of integer $k$-cycles in $X$. In a general complex $X$ two integer $k$-cycles are ''homologous'' if their difference is a linear combination with integer coefficients of boundaries of some $(k+1)$-faces. The '''integer homology group''' $H_k(X;\Z)$ is the group of homology classes of integer $k$-cycles in $X$. For a cell complex $X$ integer $k$-cycle, boundary, and integer homology group are defined analogously. (The alternative definition of the orientation is used.) </wikitex> == Alternative definition via chain complex== <wikitex>;\label{s:alt} We present the definition for $\Z_2$-coefficients. Denote by $C_k(X)=C_k(X;\Z_2)$ the set (the $\Z_2$-space) of arrangements of zeroes and units on the $s$-dimensional cells of $T$ (so $C_k(X)=\{0\}$ if there are no $k$-dimensional cells in $X$). Denote by $\partial=\partial_k \colon C_k(X)\to C_{k-1}(X)$ the extension over $C_k(X)$ of the map taking a $k$-dimensional cell $\sigma$ of $X$ to the boundary of $\sigma$. Denote $$Z_k(X)=Z_k(X;\Z_2) := \ker\partial_k\quad\text{and}\quad B_k(X)=B_k(X;\Z_2):=\mathrm{im}\partial_{k+1}.$$ Then define $H_k(X)=H_k(X;\Z_2) := Z_k(X)/B_k(X)$. </wikitex> == References == {{#RefList:}} [[Category:Definitions]] [[Category:Forgotten in Textbooks]]

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