Čech homology

The Čech homology , named after Eduard Čech , is a homology theory and therefore belongs to the mathematical branch of algebraic topology . More precisely, a sequence of groups is assigned to a topological space and a subspace contained therein . These groups , labeled, reflect properties of the topological spaces. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle {\check {H}}_{p}(E,F)}$ ${\displaystyle p=0,1,2,\ldots }$

Introductory remarks

Simplicial complexes are made up of simplices.

Historically, homology groups were initially defined for simplicial complexes , more precisely one speaks of simplicial homology . Simplicial complexes are topological spaces that are composed in a simple manner from simplices . These considerations can then be extended to topological spaces that are homeomorphic to such simplicial complexes , such spaces are called triangulable . In a further step, one would then like to define the homology groups for all topological spaces, including those that cannot be triangulated. The singular homology is such a possible generalization to all topological spaces, the Čech homology to be presented here is an alternative generalization.

In contrast to the singular homology, the homology groups are not obtained from a chain complex , but are defined directly by a Limes process. The topological space is approximated more precisely by means of overlapping by simplicial complexes and new groups are obtained from the homology groups of these simplicial complexes by means of a projective limit , which are then the Čech homology groups sought. This construction is presented in the following, after which properties and differences to singular homology are examined. The construction is based on a fixed group , the so-called coefficient group , which we suppress as far as possible. ${\displaystyle G}$

The nerve of an overlap

It is a topological space. If there is a finite cover, construct an abstract simplicial complex as follows . Let each be a corner of . A subset forms a simplex of if and only if ${\displaystyle E}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle K_{\mathcal {U}}}$ ${\displaystyle U\in {\mathcal {U}}}$ ${\displaystyle K_{\mathcal {U}}}$ ${\displaystyle \sigma \subset {\mathcal {U}}}$ ${\displaystyle K_{\mathcal {U}}}$

${\displaystyle \bigcap _{U\in \sigma }U\not =\emptyset }$.

The simplicial complex is called the nerve of overlap . Geometrically, such simplicial complexes can be realized in a Euclidean space of sufficiently large dimensions . Let yourself be guided in the following by the idea that the topological space is approximated better and better by the nerves of ever finer overlaps. The projective limits of the simplicial homology groups of these nerves will then be the Čech homology groups sought. ${\displaystyle K_{\mathcal {U}}}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle \mathbb {R} ^{d}}$

Spherical surface 6 hemispherical shells , . ${\displaystyle U_{i}^{+}}$ ${\displaystyle U_{i}^{-}}$

As an example we consider the spherical surface . If , as is usual in differential geometry , one covers with and , one obtains the abstract simplicial complex as a nerve , which corresponds to a line in a geometric realization. If you choose the finer cover from the 6 open hemispherical shells ${\displaystyle E=\{(x_{1},x_{2},x_{3})\in \mathbb {R} ^{3}\mid x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}}$ ${\displaystyle E}$ ${\displaystyle U_{1}=\{(x_{1},x_{2},x_{3})\in E\mid x_{3}<1\}}$ ${\displaystyle U_{2}=\{(x_{1},x_{2},x_{3})\in E\mid x_{3}>-1\}}$ ${\displaystyle \{\{U_{1}\},\{U_{2}\},\{U_{1},U_{2}\}\}}$

${\displaystyle U_{i}^{+}=\{(x_{1},x_{2},x_{3})\in E\mid x_{i}>0\},\quad U_{i}^{-}=\{(x_{1},x_{2},x_{3})\in E\mid x_{i}<0\},\quad i=1,2,3}$,

so the nerve is the same

${\displaystyle \{\,\,\{U_{1}^{+}\},\{U_{2}^{+}\},\{U_{3}^{+}\},\{U_{1}^{-}\},\{U_{2}^{-}\},\{U_{3}^{-}\}\,\,\}}$

${\displaystyle \cup \,\,\{\,\,\{U_{1}^{+},U_{2}^{+}\},\{U_{1}^{+},U_{2}^{+}\},\{U_{1}^{+},U_{2}^{-}\},\{U_{1}^{+},U_{3}^{-}\},\{U_{2}^{+},U_{3}^{+}\},\{U_{2}^{+},U_{1}^{-}\},\{U_{2}^{+},U_{3}^{-}\},\{U_{3}^{+},U_{1}^{-}\},\{U_{3}^{+},U_{2}^{-}\},\{U_{1}^{-},U_{2}^{-}\},\{U_{2}^{-},U_{3}^{-}\},\{U_{2}^{-},U_{3}^{-}\}\,\,\}}$

${\displaystyle \cup \,\,\{\,\,\{U_{1}^{+},U_{2}^{+},U_{3}^{+}\},\{U_{1}^{+},U_{2}^{+},U_{3}^{-}\},\{U_{1}^{+},U_{3}^{+},U_{2}^{-}\},\{U_{2}^{+},U_{3}^{+},U_{1}^{-}\},\{U_{1}^{-},U_{2}^{-},U_{3}^{+}\},\{U_{1}^{-},U_{2}^{-},U_{3}^{-}\},\{U_{1}^{-},U_{3}^{-},U_{2}^{+}\},\{U_{2}^{-},U_{3}^{-},U_{1}^{+}\}\,\,\}}$

and a geometrical realization is homeomorphic to the surface of an octahedron , which in turn is homeomorphic to the spherical surface.

Construction of the homology groups

The connection from and belongs to the partial simplex marked in red, because it also intersects , for and this does not apply, although they both intersect. ${\displaystyle U_{1}}$ ${\displaystyle U_{2}}$ ${\displaystyle U_{1}\cap U_{2}}$ ${\displaystyle F}$ ${\displaystyle U_{1}}$ ${\displaystyle U_{6}}$ ${\displaystyle F}$

To get a theory of homology we have to consider pairs of topological spaces and subspaces , where is allowed. If there is a finite cover of , then let the sub-complex of , which consists of all subsets , for which ${\displaystyle (E,F)}$ ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle F=\emptyset }$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle E}$ ${\displaystyle K'_{\mathcal {U}}}$ ${\displaystyle K_{\mathcal {U}}}$ ${\displaystyle \sigma \subset {\mathcal {U}}}$

${\displaystyle \bigcap _{U\in \sigma }U\cap F\not =\emptyset }$

is. Then there is a simplicial pair and the simplicial homology groups can be formed. We define ${\displaystyle (K_{\mathcal {U}},K'_{\mathcal {U}})}$ ${\displaystyle H_{p}(K_{\mathcal {U}},K'_{\mathcal {U}})}$

${\displaystyle {\check {H}}_{p}(E,F,{\mathcal {U}}):=H_{p}(K_{\mathcal {U}},K'_{\mathcal {U}})}$.

It should be noted that these simplicial homology groups are defined with respect to a group of coefficients mentioned above, but that they are not mentioned in this description.

Before we let the coverages become finer and finer, we have to introduce some induced mappings . Let it be a continuous mapping between pairs of topological spaces, i.e. is subspace of , subspace of , is a continuous mapping and it is . Let it now be a finite cover of . Then ${\displaystyle f:(X,Y)\rightarrow (E,F)}$ ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle f}$ ${\displaystyle X\rightarrow E}$ ${\displaystyle f(Y)\subset F}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle E}$

${\displaystyle f^{-1}({\mathcal {U}}):=\{f^{-1}(U)\mid U\in {\mathcal {U}}\}}$

an open cover of and one can also form the simplicial pair in addition to . From this we now construct a simplicial map ${\displaystyle X}$ ${\displaystyle (K_{f^{-1}({\mathcal {U}})},K'_{f^{-1}({\mathcal {U}})})}$

${\displaystyle {\tilde {f}}_{\mathcal {U}}:(K_{f^{-1}({\mathcal {U}})},K'_{f^{-1}({\mathcal {U}})})\rightarrow (K_{\mathcal {U}},K'_{\mathcal {U}})}$,

by explaining on the vertices of as follows: A vertex of is a set of the form for a generally ambiguous cover set . Choose one and define it . It can be shown that this defines a simplicial mapping which is therefore a group homomorphism ${\displaystyle {\tilde {f}}_{\mathcal {U}}}$ ${\displaystyle K_{f^{-1}({\mathcal {U}})}}$ ${\displaystyle V}$ ${\displaystyle K_{f^{-1}({\mathcal {U}})}}$ ${\displaystyle V=f^{-1}(U)}$ ${\displaystyle U\in {\mathcal {U}}}$ ${\displaystyle U}$ ${\displaystyle {\tilde {f}}_{\mathcal {U}}(V):=U}$

${\displaystyle f_{\mathcal {U}}:H_{p}(K_{f^{-1}({\mathcal {U}})},K'_{f^{-1}({\mathcal {U}})})\rightarrow H_{p}(K_{\mathcal {U}},K'_{\mathcal {U}})}$

induced between the simplicial homology groups. Furthermore, one can show that this group homomorphism no longer depends on the choices made , that is, one obtains a clear group homomorphism ${\displaystyle U\in {\mathcal {U}}}$

${\displaystyle f_{\mathcal {U}}:{\check {H}}_{p}(X,Y,f^{-1}({\mathcal {U}}))\rightarrow {\check {H}}_{p}(E,F,{\mathcal {U}})}$,

which only depends on ${\displaystyle {\mathcal {U}}}$

Now we consider a pair of topological spaces with two finite coverages and , with a finer cover, that is, for each there is a with . For each choose one and define it . One can show that a simplicial mapping is given between the simplicial by this allocation of corners actually, the course taken by the elections with dependent. As in the case of the one described above, this dependency disappears when one goes over to the homology groups, which one only receives from and dependent images ${\displaystyle (E,F)}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle V\in {\mathcal {V}}}$ ${\displaystyle U\in {\mathcal {U}}}$ ${\displaystyle V\subset U}$ ${\displaystyle V\in {\mathcal {V}}}$ ${\displaystyle U}$ ${\displaystyle {\tilde {\pi }}_{\mathcal {UV}}:(K_{\mathcal {V}},K'_{\mathcal {V}})\rightarrow K_{\mathcal {U}},K'_{\mathcal {U}}),\quad V\mapsto U}$ ${\displaystyle U\in {\mathcal {U}}}$ ${\displaystyle V\subset U}$ ${\displaystyle {\tilde {f}}_{\mathcal {U}}}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle {\mathcal {V}}}$

${\displaystyle \pi _{\mathcal {UV}}:{\check {H}}_{p}(E,F,{\mathcal {V}})\rightarrow {\check {H}}_{p}(E,F,{\mathcal {U}})}$.

For the mappings introduced here, the following relationships can be demonstrated, where a continuous mapping between pairs of topological spaces and , and finite coverages , which become finer in this order: ${\displaystyle f:(X,Y)\rightarrow (E,F)}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {W}}}$ ${\displaystyle (E,F)}$

• ${\displaystyle \pi _{\mathcal {UU}}=\mathrm {id} _{{\check {H}}_{p}(E,F,{\mathcal {U}})}}$
• ${\displaystyle \pi _{\mathcal {UW}}=\pi _{\mathcal {UV}}\circ \pi _{\mathcal {VW}}}$
• ${\displaystyle \pi _{\mathcal {UV}}\circ f_{\mathcal {V}}=f_{\mathcal {U}}\circ \pi _{\mathcal {f^{-1}(U)f^{-1}(V)}}}$

The first two equations show that the data of a projective limit are available, i.e. one can

${\displaystyle {\check {H}}_{p}(E,F):=\lim _{\leftarrow {\mathcal {U}}}({\check {H}}_{p}(E,F,{\mathcal {U}}),\pi _{\mathcal {UV}})}$

using the easily verifiable fact that the set of finite, open coverages with respect to the "fine" relation is a directed set . The third equation shows that there is a group homomorphism ${\displaystyle f_{\mathcal {U}}}$

${\displaystyle f_{*}:{\check {H}}_{p}(X,Y)\rightarrow {\check {H}}_{p}(E,F)}$

define. This applies to everyone , in the notation the dependence on is suppressed. ${\displaystyle p=0,1,2,\ldots }$ ${\displaystyle f_{*}}$ ${\displaystyle p}$

This forms the assignments

${\displaystyle (E,F)\mapsto {\check {H}}_{p}(E,F)}$

${\displaystyle f\mapsto f_{*}}$, with suppressed dependence on ${\displaystyle p}$

Functors from the category of pairs of topological spaces to the category of Abelian groups . The functor properties, that is, that the identical mapping are mapped onto the identical group homomorphisms and that the group homomorphisms of a composition match the compositions of the group homomorphisms, result directly from the corresponding properties of simplicial homology and the construction using the projective limit. These functors are called the Čech homology of the pair, the groups are called Čech homology groups. For you leave that out, that means you just write . ${\displaystyle {\check {H}}_{p}(E,F)}$ ${\displaystyle F=\emptyset }$ ${\displaystyle F}$ ${\displaystyle {\check {H}}_{p}(E)}$

properties

Many properties of the Čech homology result from the properties of the singular or simplicial homology, in which the corresponding properties of these homology theories are transferred to the projective Limes. The functor properties mentioned above show that homeomorphic pairs have the same Čech homology, since homeomorphisms between pairs apparently induce isomorphisms between the corresponding Čech homology groups. The latter are therefore topological invariants .

Comparison with singular homology

The blue area consists of arcs that move closer and closer together towards the left.

Due to the design, the Čech homology groups of simplicial pairs are isomorphic to the simplicial and therefore to the singular homology groups. In particular, applies to single-point space

${\displaystyle {\check {H}}_{p}(\{x\})={\begin{cases}G,&{\text{falls }}p=0\\0,&{\text{falls }}p>0\end{cases}}}$

Here is the coefficient group on which the construction is based, and 0, as usual, stands for the trivial group . ${\displaystyle G}$

Using homeomorphism, it immediately follows that Čech homology and singular homology match on pairs of triangulable spaces, which no longer applies to more general spaces, as the example on the right shows. This space is subspace of the plane and consists of the function graph of , and the broken line of about and up . The following applies for the singular homology , but for the Čech homology one obtains (coefficient group ), see below. ${\displaystyle E}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \textstyle ]0,{\frac {2}{\pi }}]\rightarrow \mathbb {R} }$ ${\displaystyle \textstyle x\mapsto \sin({\frac {1}{x}})}$ ${\displaystyle (0,1)}$ ${\displaystyle (0,-2)}$ ${\displaystyle \textstyle ({\frac {2}{\pi }},-2)}$ ${\displaystyle \textstyle ({\frac {2}{\pi }},1)}$ ${\displaystyle H_{1}(E)=0}$ ${\displaystyle {\check {H}}_{1}(E)=\mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$

Homotopy invariance

Two continuous functions between pairs of topological spaces are called homotop if there is a continuous mapping   with   and     for all . In this case ${\displaystyle f,g:(X,Y)\rightarrow (E,F)}$ ${\displaystyle h:(X\times [0,1],Y\times [0,1])\rightarrow (E,F)}$ ${\displaystyle h(x,0)=f(x)}$ ${\displaystyle h(x,1)=g(x)}$ ${\displaystyle x\in X}$

${\displaystyle f_{*}=g_{*}:{\check {H}}_{p}(X,Y)\rightarrow {\check {H}}_{p}(E,F)}$  for everyone . ${\displaystyle p\geq 0}$

In particular, homotopy-equivalent space pairs have isomorphic Čech homology groups.

Long homology sequence

For each pair of topological spaces and each there are homomorphisms ${\displaystyle (E,F)}$ ${\displaystyle p>0}$

${\displaystyle \partial :{\check {H}}_{p}(E,F)\rightarrow {\check {H}}_{p-1}(F)}$

(the dependency on and is suppressed) so that: ${\displaystyle p}$ ${\displaystyle (E,F)}$

If there is a continuous mapping between pairs of topological spaces, the following diagram is commutative : ${\displaystyle f:(X,Y)\rightarrow (E,F)}$

${\displaystyle {\begin{array}{ccc}{\check {H}}_{p}(X,Y)&{\xrightarrow {\partial }}&{\check {H}}_{p-1}(Y)\\\downarrow _{f_{*}}&&\downarrow _{(f|_{Y})_{*}}\\{\check {H}}_{p}(E,F)&{\xrightarrow {\partial }}&{\check {H}}_{p-1}(F)\end{array}}}$

If further and the inclusion images apply, then applies to the long homology sequence ${\displaystyle i:F\rightarrow E}$ ${\displaystyle j:E=(E,\emptyset )\rightarrow (E,F)}$

${\displaystyle \ldots \rightarrow \,\,{\check {H}}_{p}(F)\,\,{\xrightarrow {i_{*}}}\,\,{\check {H}}_{p}(E)\,\,{\xrightarrow {j_{*}}}\,\,{\check {H}}_{p}(E,F)\,\,{\xrightarrow {\partial }}\,\,{\check {H}}_{p-1}(F)\,\,\rightarrow \ldots }$,

that the composition of successive homomorphisms is the zero homomorphism, that is, the core of each homomorphism comprises the image of the previous one. Note that this property is considerably weaker than the equivalent in the singular homology, for which the long homology sequence is even exact . Finally, it should be mentioned that the homomorphisms arise from the connection homomorphisms of simplicial homology through the formation of the projective limit and that the weakness mentioned is due to the fact that the transition to the projective limit is generally not accurate. ${\displaystyle \partial }$

Clipping

Let it be a pair of topological spaces and let it be an open set whose closed shell is contained in the interior of . Then the inclusion map induces isomorphisms ${\displaystyle (E,F)}$ ${\displaystyle A\subset E}$ ${\displaystyle F}$ ${\displaystyle i:(E\setminus A,F\setminus A)\rightarrow (E,F)}$

${\displaystyle i_{*}:{\check {H}}_{p}(E\setminus A,F\setminus A)\rightarrow {\check {H}}_{p}(E,F)}$  for all ${\displaystyle p\geq 0}$

This is called the cutting property, because it is imagined that you have cut out of the couple . Note that the corresponding property of singular homology holds without the openness of . This requirement cannot be dispensed with in the Čech homology ${\displaystyle A}$ ${\displaystyle (E,F)}$ ${\displaystyle A}$

continuity

A special feature of the Čech homology, which is missing in the singular homology, is the so-called continuity, which is based on the construction of the projective limes. Let it be a directed set. For each, let a pair of compact spaces be , that is, and both are compact, and for each in let be a continuous mapping, so that the following relationships hold: ${\displaystyle I}$ ${\displaystyle \alpha \in I}$ ${\displaystyle (E_{\alpha },F_{\alpha })}$ ${\displaystyle E_{\alpha }}$ ${\displaystyle F_{\alpha }}$ ${\displaystyle \alpha \leq \beta }$ ${\displaystyle I}$ ${\displaystyle f_{\alpha ,\beta }:(E_{\beta },F_{\beta })\rightarrow (E_{\alpha },F_{\alpha })}$

${\displaystyle f_{\alpha ,\alpha }=\mathrm {id} _{(E_{\alpha },F_{\alpha })}}$   for all ${\displaystyle \alpha \in I}$

${\displaystyle f_{\alpha ,\gamma }=f_{\alpha ,\beta }\circ f_{\beta ,\gamma }}$   for everyone with . ${\displaystyle \alpha ,\beta ,\gamma \in I}$ ${\displaystyle \alpha \leq \beta \leq \gamma }$

With this data one can on the one hand the projective Limes

${\displaystyle (E,F):=\lim _{\leftarrow \alpha \in I}((E_{\alpha },F_{\alpha }),f_{\alpha ,\beta })}$

of the compact pairs, on the other hand, by using the functor of the -th Čech homology, the data and for in , from which the projective limit of the Čech homology groups can be formed. The continuity property says that the expected relationship holds: ${\displaystyle p}$ ${\displaystyle {\check {H}}_{p}(E_{\alpha },F_{\alpha })}$ ${\displaystyle f_{{\alpha ,\beta }*}}$ ${\displaystyle \alpha \leq \beta }$ ${\displaystyle I}$

${\displaystyle {\check {H}}_{p}(\lim _{\leftarrow \alpha \in I}((E_{\alpha },F_{\alpha }),f_{\alpha ,\beta }))\cong \lim _{\leftarrow \alpha \in I}({\check {H}}_{p}(E_{\alpha },F_{\alpha }),f_{{\alpha ,\beta }*})}$.

example

Consider as an application to space , of the above means of was constructed Graphs, the spaces resulting from by adding the full rectangle with the corners , , and emerge. If you let the left side of the added rectangles move steadily towards the right, you can see that homotop is a simply closed line and thus a circle. ${\displaystyle E}$ ${\displaystyle \textstyle \sin({\frac {1}{x}})}$ ${\displaystyle E_{n}}$ ${\displaystyle E}$ ${\displaystyle (0,1)}$ ${\displaystyle (0,-2)}$ ${\displaystyle ({\tfrac {1}{n}},-2)}$ ${\displaystyle ({\tfrac {1}{n}},1)}$ ${\displaystyle E_{n}}$

The rooms are homotopic to a circular line. ${\displaystyle E_{n}}$

So, because of the homotopy invariance and agreement of singular homology and Čech homology for triangulable spaces , we consider the coefficient group again. For is the inclusion figure . Then the identity and continuity supplies ${\displaystyle {\check {H}}_{1}(E_{n})\cong H_{1}(E_{n})\cong \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle n\leq m}$ ${\displaystyle f_{n,m}}$ ${\displaystyle E_{m}\rightarrow E_{n}}$ ${\displaystyle f_{n,m*}}$

${\displaystyle {\check {H}}_{1}(E)={\check {H}}_{1}(\bigcap _{n\in \mathbb {N} }E_{n})={\check {H}}_{1}(\lim _{\leftarrow n\in \mathbb {N} }(E_{n},f_{n,m}))\cong \lim _{\leftarrow n\in \mathbb {N} }({\check {H}}_{1}(E_{n}),f_{n,m*})\cong \mathbb {Z} }$.

The singular homology, on the other hand, is 0. This is essentially due to the fact, without going into details, that no simply closed curve can "go around" space. The space is therefore an example of a space that cannot be triangulated, because in the case of triangulation, singular homology and Čech homology would have to match, which is not the case here. ${\displaystyle H_{1}(E)}$ ${\displaystyle E}$

literature

• Andrew Wallace: Algebraic Topology, Homology and Cohomology , WA Benjamin Inc. (1969)