Simplicial homology

In algebraic topology , a branch of mathematics , simplicial homology is a method that assigns a sequence of Abelian groups to any simplicial complex . Vividly speaking, it counts the holes of different dimensions in the underlying space.

Simplicial complexes

A simplicial complex is a set of simplices (clearly determined by their corner points) , so that each side face of one of the simplices is again in this set. Simple examples are polygons and polyhedra . According to a theorem of topology , one can triangulate every differentiable manifold, i.e. understand it as a simplicial complex (SK). ${\displaystyle K}$

Simplicial homology

For a simplicial complex we consider for the free Abelian group over the set of -simplices of the simplicial complex . ${\displaystyle K}$ ${\displaystyle n=0,1,2,\ldots }$ ${\displaystyle n}$ ${\displaystyle C_{n}(K)}$

Elements of are thus formal sums of form ${\displaystyle C_{n}(K)}$

${\displaystyle \sum _{i=1}^{r}a_{i}\sigma _{i}}$

with and a simplex of . It is required that the following applies if the simplices and have the opposite orientation. ${\displaystyle a_{i}\in \mathbb {Z} }$ ${\displaystyle \sigma _{i}}$ ${\displaystyle n}$ ${\displaystyle K}$ ${\displaystyle \sigma _{i}=-\sigma _{j}}$ ${\displaystyle \sigma _{i}}$ ${\displaystyle \sigma _{j}}$

The "edge map" maps each simplex to the alternating sum of its side surfaces, that is ${\displaystyle \partial \colon C_{n}(K)\rightarrow C_{n-1}(K)}$

${\displaystyle \partial ([v_{0},\dots ,v_{n}]):=\sum _{i=0}^{n}(-1)^{i}[v_{0},\ldots ,{\hat {v_{i}}},\ldots ,v_{n}]\,,}$

where means that it is left out. The alternating sign factors can also be interpreted as "geometric orientation numbers". ${\displaystyle {\hat {v}}_{i}}$ ${\displaystyle v_{i}}$

This edge mapping, which is defined on the generators , continues through linear continuation ${\displaystyle C_{n}(K)}$

${\displaystyle \partial (\sum _{i=1}^{r}a_{i}\sigma _{i})=\sum _{i=1}^{r}a_{i}\partial (\sigma _{i})}$

clearly proceeded to a figure . It's easy to calculate that ${\displaystyle \partial :C_{n}(K)\rightarrow C_{n-1}(K)}$

${\displaystyle \partial \circ \partial =0}$

applies. so is a chain complex . ${\displaystyle (C_{*}(K),\partial )}$

The homology of this chain complex is called the simplicial homology of ${\displaystyle K}$ and is denoted by. ${\displaystyle H_{*}(K)}$

example

Calculation example

triangle

We want to calculate the homology groups of the triangle (consisting of three 0-simplices and the three connecting 1-simplices, no 2-simplex and no higher-dimensional simplices). ${\displaystyle v_{1},v_{2},v_{3}}$

According to the definition of the boundary operator , then: ${\displaystyle \partial _{0}([v_{i}])=0}$

${\displaystyle \mathrm {ker} (\partial _{0})=C_{0}=\{a_{1}[v_{1}]+a_{2}[v_{2}]+a_{3}[v_{3}]|a_{1},a_{2},a_{3}\in \mathbb {Z} \}\cong \mathbb {Z} \oplus \mathbb {Z} \oplus \mathbb {Z} }$

d. H. all 0 chains are in the core.

For a 1 chain is ${\displaystyle c_{1}=b_{1}[v_{1},v_{2}]+b_{2}[v_{2},v_{3}]+b_{3}[v_{3},v_{1}]}$

${\displaystyle \partial _{1}(c_{1})=(b_{3}-b_{1})[v_{1}]+(b_{1}-b_{2})[v_{2}]+(b_{2}-b_{3})[v_{3}]}$

From this you get

${\displaystyle \mathrm {Im} (\partial _{1})=\{(b_{3}-b_{1})[v_{1}]+(b_{1}-b_{2})[v_{2}]+(b_{2}-b_{3})[v_{3}]|b_{1},b_{2},b_{3}\in \mathbb {Z} \}}$,

a 0-chain belongs to the picture of if and only if ${\displaystyle c_{0}=a_{1}[v_{1}]+a_{2}[v_{2}]+a_{3}[v_{3}]}$ ${\displaystyle \partial _{1}}$

${\displaystyle a_{1}=b_{3}-b_{1}}$

${\displaystyle a_{2}=b_{1}-b_{2}}$

${\displaystyle a_{3}=b_{2}-b_{3}}$,

so exactly when . It follows ${\displaystyle a_{1}+a_{2}+a_{3}=0}$

${\displaystyle H_{0}(S)\cong (\mathbb {Z} \oplus \mathbb {Z} \oplus \mathbb {Z} )/(a_{1}+a_{2}+a_{3}=0)\cong \mathbb {Z} }$

To calculate the first homology group: For a 1 chain

${\displaystyle c_{1}=b_{1}[v_{1},v_{2}]+b_{2}[v_{2},v_{3}]+b_{3}[v_{3},v_{1}]}$

is if and only if , so ${\displaystyle \partial _{1}(c_{1})=0}$ ${\displaystyle b_{1}=b_{2}=b_{3}}$

${\displaystyle \mathrm {ker} (\partial _{1})=\{b[v_{1},v_{2}]+b[v_{2},v_{3}]+b[v_{3},v_{1}]|b\in \mathbb {Z} \}\cong \mathbb {Z} .}$

Because there is no 2-simplices, core and picture are trivial . With this we get: ${\displaystyle \partial _{2}}$ ${\displaystyle \mathrm {ker} (\partial _{2})=\mathrm {Im} (\partial _{2})=0}$

${\displaystyle H_{1}(S)=\mathrm {ker} (\partial _{1})/\mathrm {Im} (\partial _{2})=\mathrm {ker} (\partial _{1})\cong \mathbb {Z} }$

${\displaystyle H_{2}(S)=\mathrm {ker} (\partial _{2})/\mathrm {Im} (\partial _{3})\cong 0}$

and trivially for everyone . ${\displaystyle H_{i}(S)\cong 0}$ ${\displaystyle i>2}$

Further examples

The following apply:

• Is the simplicial complex that triangulates the triangle with content. That means the complex as above, only with the addition of the 2-simplex. Then it arises ${\displaystyle K}$ ${\displaystyle H_{0}(K)=\mathbb {Z} ,\,H_{1}(K)=H_{2}(K)=\ldots =0}$
• For the 2-torus is valid and for . ${\displaystyle T=S^{1}\times S^{1}}$ ${\displaystyle H_{1}(T)=\mathbb {Z} ^{2},\,H_{2}(T)=\mathbb {Z} }$ ${\displaystyle H_{i}(T)=0}$ ${\displaystyle i>2}$
• The same applies to the Klein bottle and to . ${\displaystyle K}$ ${\displaystyle H_{1}(K)=\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle H_{i}(K)=0}$ ${\displaystyle i\geq 2}$
• It applies and for all . ${\displaystyle H_{1}(\mathbb {R} P^{2})=\mathbb {Z} }$ ${\displaystyle H_{i}(\mathbb {R} P^{2})=0}$ ${\displaystyle i\geq 2}$
• Let it be a simplicial complex with connected components . ${\displaystyle K}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle H_{0}(K)=\mathbb {Z} ^{n}}$

Functoriality

Simplicial illustrations

A simplicial mapping induces a chain mapping ${\displaystyle f\colon K\to L}$

${\displaystyle f_{*}:C_{*}(K)\rightarrow C_{*}(L)}$

by

${\displaystyle f_{*}(\sum _{i=1}^{r}a_{i}\sigma _{i})=\sum _{i=1}^{r}a_{i}f(\sigma _{i})}$

and because of a well-defined mapping ${\displaystyle df=fd}$

${\displaystyle f_{*}:H_{*}(K)\rightarrow H_{*}(L)}$.

Continuous images

Be

${\displaystyle f:\vert K\vert \to \vert L\vert }$

a continuous mapping between the geometric realizations of two simplicial complexes and . We denote with the barycentric subdivision of and with the -fold iterated barycentric subdivision. It applies . ${\displaystyle K}$ ${\displaystyle L}$ ${\displaystyle Bd(K)}$ ${\displaystyle K}$ ${\displaystyle Bd^{n}(K)}$ ${\displaystyle n}$ ${\displaystyle \vert Bd^{n}(K)\vert =\vert K\vert }$

According to the simplicial approximation theorem, there is a , so that a simplicial approximation ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle f:\vert Bd^{n}(K)\vert \to \vert L\vert }$

${\displaystyle g:Bd^{n}(K)\rightarrow L}$

owns.

Then it will be

${\displaystyle f_{*}:H_{*}(K)\rightarrow H_{*}(L)}$

defined as the connection of with canonical isomorphism . One can show that the homomorphism defined in this way is independent of the choice of simplicial approximation. ${\displaystyle g_{*}}$ ${\displaystyle H_{*}(K)\to H_{*}(Bd^{n}(K))}$ ${\displaystyle f_{*}}$

Simplicial homology with coefficients

For an Abelian group and a simplicial complex one defines ${\displaystyle G}$ ${\displaystyle K}$

${\displaystyle C_{*}(K,G)=C_{*}(K)\otimes _{\mathbb {Z} }G}$,

Elements of are thus formal sums of the form with and a simplex in . The boundary operator continues with ${\displaystyle C_{n}(K,G)}$ ${\displaystyle \textstyle \sum _{i=1}^{r}a_{i}\sigma _{i}}$ ${\displaystyle a_{i}\in G}$ ${\displaystyle \sigma _{i}}$ ${\displaystyle n}$ ${\displaystyle K}$

${\displaystyle d(\sum _{i=1}^{r}a_{i}\sigma _{i})=\sum _{i=1}^{r}a_{i}d\sigma _{i}}$.

The homology with coefficients in G

${\displaystyle H_{*}(X,G)}$

is defined as the homology of the chain complex . ${\displaystyle (C_{*}(X,G),d)}$

Simplicial versus Singular Homology

The simplicial homology of a simplicial complex is isomorphic to the singular homology of its geometric realization:

${\displaystyle H_{*}(K,G)=H_{*}(\vert K\vert ,G)}$.

literature

• Stocker, Ralph; Zieschang, Heiner: Algebraic Topology. An introduction. 2nd Edition. Math guides. BG Teubner, Stuttgart, 1994. ISBN 3-519-12226-X .