Homology theory

The theory of homology ( dr. Greek ὁμός “equal, identical; common; mutual” and λόγος “doctrine, science ”) is a branch of mathematics that studies the constructions of some topological invariants called homology groups and cohomology groups . Also called homology theories are specific constructions of homology groups.

In the simplest case, a topological space $X$ ${\ displaystyle X}$ $X$ the sequence of abelian homology groups is compared $H_{k}(X)$ ${\ displaystyle H_ {k} (X)}$ $H_ {k} (X)$ numbered by natural numbers $k$ ${\ displaystyle k}$ $k$ . They are homotopy invariants and, in contrast to homotopy groups , they are easier to calculate and more geometrically clear, but for simply connected spaces they carry the same amount of information ^[1] .

However, the definition of homology is less explicit and uses some technical machinery ^[2] , and therefore there are several different theories of homology - both defined only for “good” topological spaces or requiring an additional structure , and more complex ones designed to work with pathological examples. Nevertheless, with the exception of such pathological cases, they usually coincide: for cell spaces this is ensured by the Steenrod – Eilenberg axioms .

Other common concepts of homology theory are homology $H_{k}(X,A)$ ${\ displaystyle H_ {k} (X, A)}$ ${\ displaystyle H_ {k} (X, A)}$ with coefficients in an abelian group $A$ ${\ displaystyle A}$ $A$ relative homology $H_{k}(X,Y)$ ${\ displaystyle H_ {k} (X, Y)}$ ${\ displaystyle H_ {k} (X, Y)}$ pairs of spaces $X\supset Y$ ${\ displaystyle X \ supset Y}$ ${\ displaystyle X \ supset Y}$ and cohomology $H^{k}(X)$ ${\ displaystyle H ^ {k} (X)}$ ${\ displaystyle H ^ {k} (X)}$ whose definitions are, in a sense, dual to the definition of homology. Cohomology is often considered because of the presence of multiplication on them. $H^{k}(X)\otimes H^{l}(X)\to H^{k+l}(X)$ ${\ displaystyle H ^ {k} (X) \ otimes H ^ {l} (X) \ to H ^ {k + l} (X)}$ ${\ displaystyle H ^ {k} (X) \ otimes H ^ {l} (X) \ to H ^ {k + l} (X)}$ turning them into graded algebra .

Also called cohomology are invariants associated with other mathematical objects - groups , Lie algebras , sheaves . They are united by formal similarity - for example, the presence in their definition of the concept of homology of a chain complex - and in some cases, the presence of constructions that map topological spaces with suitable homologies to such objects.

General definition

Recall that $k$ ${\ displaystyle k}$ $k$ Homotopy group $\pi _{k}(X)$ ${\ displaystyle \ pi _ {k} (X)}$ $\pi _{k}(X)$ of space $X$ ${\ displaystyle X}$ $X$ Is a lot of mappings from $k$ ${\ displaystyle k}$ $k$ -dimensional sphere in $X$ ${\ displaystyle X}$ $X$ considered up to a continuous deformation . To determine homology $H_{k}(X)$ ${\ displaystyle H_ {k} (X)}$ $H_{k}(X)$ mappings of spheres are replaced by $k$ ${\ displaystyle k}$ $k$ -cycles that intuitively represent as closed (that is, without boundaries) oriented films of dimension $k$ ${\ displaystyle k}$ $k$ inside $X$ ${\ displaystyle X}$ $X$ , but in different definitions they formalize in different ways. The condition of continuous deformability is replaced by the condition that the difference of the cycles (their union, in which the second is taken with the opposite orientation) is the oriented boundary of the cycle of dimension one greater.

In standard notation, the group $k$ ${\ displaystyle k}$ -cycles - $Z_{k}(X)$ ${\ displaystyle Z_ {k} (X)}$ (from German. Zyklus - "cycle"), group $k$ ${\ displaystyle k}$ -boundaries - $B_{k}(X)$ ${\ displaystyle B_ {k} (X)}$ (from the English boundary - “border”), and the phrase “homology is a cycle up to a boundary” is written as

H_{k}(X)=Z_{k}(X)/B_{k}(X)

{\ displaystyle H_ {k} (X) = Z_ {k} (X) / B_ {k} (X)}

.

To formalize this idea, it is necessary to strictly define the cycles and their boundaries, which for cycles of dimension $k>2$ ${\ displaystyle k> 2}$ leads to some difficulties ^[1] . The solution is to define an intermediate concept of a group $k$ ${\ displaystyle k}$ -chains $C_{k}(X)$ ${\ displaystyle C_ {k} (X)}$ consisting of formal linear combinations of mappings in $X$ ${\ displaystyle X}$ some standard elements depending on the chosen design. The border of standard elements is defined as a linear combination of standard elements of dimension one less with suitable orientations, which induces the display of the border $\partial _{k}:C_{k}(X)\to C_{k-1}(X)$ ${\ displaystyle \ partial _ {k}: C_ {k} (X) \ to C_ {k-1} (X)}$ . Then $k$ ${\ displaystyle k}$ -cycles are defined as $k$ ${\ displaystyle k}$ -chains with a zero border (in order for the equality of the border to zero to make sense, it is necessary to take not only positive, but also any linear combinations of standard elements, and set the border display with a sign). Thus, loops are the core , and borders are the image of the border:

Z_{k}(X)=Ker(\partial _{k}:C_{k}(X)\to C_{k-1}(X)),~~~~B_{k}(X)=Im(\partial _{k+1}:C_{k+1}(X)\to C_{k}(X))

{\ displaystyle Z_ {k} (X) = Ker (\ partial _ {k}: C_ {k} (X) \ to C_ {k-1} (X)), ~~~~ B_ {k} (X ) = Im (\ partial _ {k + 1}: C_ {k + 1} (X) \ to C_ {k} (X))}

.

The condition that all boundaries are cycles takes the form of a chain complex condition: $\partial _{k+1}\circ \partial _{k}=0$ ${\ displaystyle \ partial _ {k + 1} \ circ \ partial _ {k} = 0}$ , and the homology of a topological space are the homology of this complex.

The choice of standard elements and the display of the border differs depending on the theory. In the theory of singular homology, such elements are simplexes , and a boundary mapping associates a simplex with the alternating sum of its faces. In the theory of simplicial homologies defined for simplicial complexes , there are also simplices, but not all, but included in the chosen simplicial decomposition. In the theory of cell homology defined for a cell complex , these are hyperspheres from a suitable skeleton, and the mapping of the boundary is more complicated.

Homological Theories

Simplicial homology - homology is defined for very simple spaces ( simplicial complexes ).

They are defined quite simply, but the proof of their invariance and functoriality is rather complicated.

Singular homology is another homology theory proposed by Lefschetz . Their definition requires working with infinite-dimensional spaces, but invariance and functoriality immediately become obvious.

Cech homology is a homology theory that is best suited for working with pathological spaces.

Homology with coefficients in arbitrary groups

One can define homology, allowing the coefficients of simplexes in chains to be elements of any abelian group $G$ ${\ displaystyle G}$ . That is, instead of groups $C_{k}(X)$ ${\ displaystyle C_ {k} (X)}$ view groups $C_{k}(X)\otimes G$ ${\ displaystyle C_ {k} (X) \ otimes G}$ .

Homology groups (simplicial, singular, etc.) spaces $X$ ${\ displaystyle X}$ with odds in the group $G$ ${\ displaystyle G}$ are designated $H_{k}(X;G).$ ${\ displaystyle H_ {k} (X; G).}$ Usually apply a group of real numbers $\mathbb {R}$ ${\ displaystyle \ mathbb {R}}$ rational numbers $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ , or a cyclic group of residues modulo $m$ ${\ displaystyle m}$ - $\mathbb {Z} _{m}$ ${\ displaystyle \ mathbb {Z} _ {m}}$ , and usually taken $m=p$ ${\ displaystyle m = p}$ Is a prime, then $\mathbb {Z} _{p}$ ${\ displaystyle \ mathbb {Z} _ {p}}$ is a field .

Another description. Applying to the complex $C_{*}(X)$ ${\ displaystyle C _ {*} (X)}$

\ldots {\xleftarrow {}}C_{n-1}(X){\xleftarrow {}}C_{n}(X){\xleftarrow {}}C_{n+1}(X){\xleftarrow {}}\ldots

{\ displaystyle \ ldots {\ xleftarrow {}} C_ {n-1} (X) {\ xleftarrow {}} C_ {n} (X) {\ xleftarrow {}} C_ {n + 1} (X) {\ xleftarrow {}} \ ldots}

functor $\cdot \otimes G$ ${\ displaystyle \ cdot \ otimes G}$ we get the complex

\ldots {\xleftarrow {}}C_{n-1}(X)\otimes G{\xleftarrow {}}C_{n}(X)\otimes G{\xleftarrow {}}C_{n+1}(X)\otimes G{\xleftarrow {}}\ldots

{\ displaystyle \ ldots {\ xleftarrow {}} C_ {n-1} (X) \ otimes G {\ xleftarrow {}} C_ {n} (X) \ otimes G {\ xleftarrow {}} C_ {n + 1 } (X) \ otimes G {\ xleftarrow {}} \ ldots}

,

whose homology is homology with coefficients in $G$ ${\ displaystyle G}$ .

Cohomology

In addition to chains, one can introduce the concept of cochains - mappings of the vector space of chains into a group $G$ ${\ displaystyle G}$ . That is, the cochain space $C^{k}(X)=\operatorname {Hom} (C_{k}(X),G)$ ${\ displaystyle C ^ {k} (X) = \ operatorname {Hom} (C_ {k} (X), G)}$ .

Boundary operator $\delta ^{k}:C^{k}\to C^{k+1}$ ${\ displaystyle \ delta ^ {k}: C ^ {k} \ to C ^ {k + 1}}$ determined by the formula: $(\delta ^{k}x)(c)=x(d_{k+1}c)$ ${\ displaystyle (\ delta ^ {k} x) (c) = x (d_ {k + 1} c)}$ (Where $x\in C^{k},\;c\in C_{k+1}$ ${\ displaystyle x \ in C ^ {k}, \; c \ in C_ {k + 1}}$ ) For such a boundary operator,

\delta ^{k+1}\delta ^{k}=0

{\ displaystyle \ delta ^ {k + 1} \ delta ^ {k} = 0}

, namely

(\delta ^{k+1}\delta ^{k}(x))(c)=\delta ^{k}x(d_{k+2}c)=x(d_{k+1}d_{k+2}c)=x(0)=0

{\ displaystyle (\ delta ^ {k + 1} \ delta ^ {k} (x)) (c) = \ delta ^ {k} x (d_ {k + 2} c) = x (d_ {k + 1 } d_ {k + 2} c) = x (0) = 0}

.

Therefore, similarly to what was said above, we can introduce the concept of cocycles $Z^{k}(X,G)=\operatorname {Ker} \delta ^{k}$ ${\ displaystyle Z ^ {k} (X, G) = \ operatorname {Ker} \ delta ^ {k}}$ , borders $B^{k}(X,G)=\operatorname {Im} \delta ^{k-1}$ ${\ displaystyle B ^ {k} (X, G) = \ operatorname {Im} \ delta ^ {k-1}}$ and cohomology $H^{k}(X,G)=Z^{k}(X,G)/B^{k}(X,G)$ ${\ displaystyle H ^ {k} (X, G) = Z ^ {k} (X, G) / B ^ {k} (X, G)}$ .

The concept of cohomology is dual to the concept of homology.

If a $G$ ${\ displaystyle G}$ Is a ring then in the cohomology group $H^{*}(X,G)$ ${\ displaystyle H ^ {*} (X, G)}$ natural multiplication is defined (the work of Kolmogorov - Alexander or $\cup$ ${\ displaystyle \ cup}$ -nproduct), which turns this group into a graded ring , called the cohomology ring .

In the case when $X$ ${\ displaystyle X}$ - differentiable manifold , cohomology ring $H^{*}(X,\mathbb {R} )$ ${\ displaystyle H ^ {*} (X, \ mathbb {R})}$ can be calculated using differential forms on $X$ ${\ displaystyle X}$ (see Theorem de Rama ).

The concept of cohomology was introduced by Alexander and Kolmogorov .

Relative homology and exact homology sequence

Take the case of two topological spaces $Y\subset X$ ${\ displaystyle Y \ subset X}$ . Chain group $C_{k}(Y)\subset C_{k}(X)$ ${\ displaystyle C_ {k} (Y) \ subset C_ {k} (X)}$ (chains can be either with integer coefficients or with coefficients in any group $G$ ${\ displaystyle G}$ ) Relative chains will be called elements of the factor group $C_{k}(X,Y)=C_{k}(X)/C_{k}(Y)$ ${\ displaystyle C_ {k} (X, Y) = C_ {k} (X) / C_ {k} (Y)}$ . Since the boundary operator $d$ ${\ displaystyle d}$ on the subspace homology group $Y$ ${\ displaystyle Y}$ translates $d_{k}\colon C_{k}(Y)\to C_{k-1}(Y)$ ${\ displaystyle d_ {k} \ colon C_ {k} (Y) \ to C_ {k-1} (Y)}$ , then can be determined on the quotient group $C_{k}(X,Y)$ ${\ displaystyle C_ {k} (X, Y)}$ boundary operator (we denote it in the same way) $d_{k}\colon C_{k}(X,Y)\to C_{k-1}(X,Y)$ ${\ displaystyle d_ {k} \ colon C_ {k} (X, Y) \ to C_ {k-1} (X, Y)}$ .

Those relative chains that the boundary operator translates into $0$ ${\ displaystyle 0}$ will be called relative cycles $Z_{k}(X,Y)$ ${\ displaystyle Z_ {k} (X, Y)}$ , and the chains that are its values are relative boundaries $B_{k}(X,Y)$ ${\ displaystyle B_ {k} (X, Y)}$ . Because $dd=0$ ${\ displaystyle dd = 0}$ on absolute chains, then the same will be true for relative, hence $B_{k}(X,Y)\subset Z_{k}(X,Y)$ ${\ displaystyle B_ {k} (X, Y) \ subset Z_ {k} (X, Y)}$ . Factor group $H_{k}(X,Y)=Z_{k}(X,Y)/B_{k}(X,Y)$ ${\ displaystyle H_ {k} (X, Y) = Z_ {k} (X, Y) / B_ {k} (X, Y)}$ called a group of relative homologies .

Since every absolute cycle in $H_{k}(X)$ ${\ displaystyle H_ {k} (X)}$ is also relative, then we have a homomorphism $j_{k}:H_{k}(X)\to H_{k}(X,Y)$ ${\ displaystyle j_ {k}: H_ {k} (X) \ to H_ {k} (X, Y)}$ By the functorial property, the embedding $i_{k}:Y\to X$ ${\ displaystyle i_ {k}: Y \ to X}$ leads to homomorphism $i_{*}:H_{k}(Y)\to H_{k}(X)$ ${\ displaystyle i _ {*}: H_ {k} (Y) \ to H_ {k} (X)}$ .

In turn, we can construct a homomorphism $d_{*k}:H_{k}(X,Y)\to H_{k-1}(Y)$ ${\ displaystyle d _ {* k}: H_ {k} (X, Y) \ to H_ {k-1} (Y)}$ which we define as follows. Let be $c_{k}\in C_{k}(X,Y)$ ${\ displaystyle c_ {k} \ in C_ {k} (X, Y)}$ - a relative chain that defines a cycle from $H_{k}(X,Y)$ ${\ displaystyle H_ {k} (X, Y)}$ . Consider it as an absolute chain in $C_{k}(X)$ ${\ displaystyle C_ {k} (X)}$ (accurate to elements $C_{k}(Y)$ ${\ displaystyle C_ {k} (Y)}$ ) Since this is a relative cycle, then $d_{k}c$ ${\ displaystyle d_ {k} c}$ will be zero up to a certain chain $c_{k-1}\in C_{k-1}(Y)$ ${\ displaystyle c_ {k-1} \ in C_ {k-1} (Y)}$ . Put $d_{*k}$ ${\ displaystyle d _ {* k}}$ equal to the chain homology class $c_{k-1}=d_{k}c\in Z_{k-1}(Y)$ ${\ displaystyle c_ {k-1} = d_ {k} c \ in Z_ {k-1} (Y)}$ .

If we take another absolute chain $c'_{k}\in C_{k}(X)$ ${\ displaystyle c '_ {k} \ in C_ {k} (X)}$ defining the same relative cycle, then we will have $c=c'+u$ ${\ displaystyle c = c '+ u}$ where $u\in C_{k}(Y)$ ${\ displaystyle u \ in C_ {k} (Y)}$ . We have $d_{k}c=d_{k}c'+d_{k}u$ ${\ displaystyle d_ {k} c = d_ {k} c '+ d_ {k} u}$ , but since $d_{k}u$ ${\ displaystyle d_ {k} u}$ is the border in $Z_{k-1}(Y)$ ${\ displaystyle Z_ {k-1} (Y)}$ then $d_{k}c$ ${\ displaystyle d_ {k} c}$ and $d_{k}c'$ ${\ displaystyle d_ {k} c '}$ define the same element in the homology group $H_{k-1}(Y)$ ${\ displaystyle H_ {k-1} (Y)}$ . If we take a different relative cycle $c^{″}$ ${\ displaystyle c ''}$ giving the same element in the group of relative homologies $c=c''+b$ ${\ displaystyle c = c '' + b}$ where $b$ ${\ displaystyle b}$ Is the relative boundary, due to the fact that $b$ ${\ displaystyle b}$ boundary for relative homologies $b=d_{k+1}x+v$ ${\ displaystyle b = d_ {k + 1} x + v}$ where $v\in C_{k}(Y)$ ${\ displaystyle v \ in C_ {k} (Y)}$ from here $d_{k}c=d_{k}c''+d_{k}d_{k+1}x+d_{k}v$ ${\ displaystyle d_ {k} c = d_ {k} c '' + d_ {k} d_ {k + 1} x + d_ {k} v}$ but $dd=0$ ${\ displaystyle dd = 0}$ , but $d_{k}v$ ${\ displaystyle d_ {k} v}$ - border in $Z_{k-1}(Y)$ ${\ displaystyle Z_ {k-1} (Y)}$ .

Therefore, the homology class $d_{*k}c_{k}$ ${\ displaystyle d _ {* k} c_ {k}}$ defined uniquely. Clearly linear operator $d_{*k}$ ${\ displaystyle d _ {* k}}$ that he is a homomorphism. So we have homomorphisms:

i_{*k}\colon H_{k}(Y)\to H_{k}(X)

{\ displaystyle i _ {* k} \ colon H_ {k} (Y) \ to H_ {k} (X)}

;

j_{*k}\colon H_{k}(X)\to H_{k}(X,Y)

{\ displaystyle j _ {* k} \ colon H_ {k} (X) \ to H_ {k} (X, Y)}

and

d_{*k}\colon H_{k}(X,Y)\to H_{k-1}(Y)

{\ displaystyle d _ {* k} \ colon H_ {k} (X, Y) \ to H_ {k-1} (Y)}

;

...\to H_{k}(Y)\to H_{k}(X)\to H_{k}(X,Y)\to H_{k-1}(Y)\to ...

{\ displaystyle ... \ to H_ {k} (Y) \ to H_ {k} (X) \ to H_ {k} (X, Y) \ to H_ {k-1} (Y) \ to .. .}

It can be proved that this sequence is exact , that is, the image of any homomorphism is equal to the kernel of the next homomorphism.

Axioms of Steenrod - Eilenberg

In addition to the simplicial and singular homology already known to us, there are other theories of homology and cohomology, for example, cellular homology , Aleksandrov-Cech cohomology, de Ram cohomology , etc. Stinrod and Eilenberg defined a system of axioms of the theory of (co) homology. First, they determine the so-called. valid pair class $D$ ${\ displaystyle D}$ topological spaces satisfying the following properties:

If a $(X,Y)\in D,$ ${\ displaystyle (X, Y) \ in D,}$ then $(X,X)\in D,$ ${\ displaystyle (X, X) \ in D,}$ $(X,\varnothing )\in D,$ ${\ displaystyle (X, \ varnothing) \ in D,}$ $(Y,Y)\in D$ ${\ displaystyle (Y, Y) \ in D}$ and $(Y,\varnothing )\in D$ ${\ displaystyle (Y, \ varnothing) \ in D}$ .
If a $(X,Y)\in D$ ${\ displaystyle (X, Y) \ in D}$ then $(X\times I,Y\times I)\in D$ ${\ displaystyle (X \ times I, Y \ times I) \ in D}$ where $I$ ${\ displaystyle I}$ - closed interval [0,1].
$(*,\varnothing )\in D$ ${\ displaystyle (*, \ varnothing) \ in D}$ where $*$ ${\ displaystyle *}$ - single point space.

In the Steenrod - Eilenberg homology theory, each admissible pair and any integer k corresponds to an Abelian group $H_{k}(X,Y)$ ${\ displaystyle H_ {k} (X, Y)}$ and continuous display of pairs $f\colon (X,Y)\to (X',Y')$ ${\ displaystyle f \ colon (X, Y) \ to (X ', Y')}$ homomorphism corresponds $f_{*k}\colon H_{k}(X,Y)\to H_{k}(X',Y')$ ${\ displaystyle f _ {* k} \ colon H_ {k} (X, Y) \ to H_ {k} (X ', Y')}$ (Space $X$ ${\ displaystyle X}$ identified with a couple $(X,\varnothing )$ ${\ displaystyle (X, \ varnothing)}$ ) , and $H_{k}(X)$ ${\ displaystyle H_ {k} (X)}$ with $H_{k}(X,\varnothing )$ ${\ displaystyle H_ {k} (X, \ varnothing)}$ ), and the following axioms are fulfilled:

The identity mapping of the pair $i d$ ${\ displaystyle id}$ corresponds to an identical homomorphism $id_{*k}$ ${\ displaystyle id _ {* k}}$ .
$(gf)_{*k}=g_{*k}f_{*k}$ ${\ displaystyle (gf) _ {* k} = g _ {* k} f _ {* k}}$ ( functoriality )
A boundary homomorphism is defined $d_{*k}\colon H_{k}(X,Y)\to H_{k-1}(Y)$ ${\ displaystyle d _ {* k} \ colon H_ {k} (X, Y) \ to H_ {k-1} (Y)}$ , and if $f\colon (X,Y)\to (X',Y')$ ${\ displaystyle f \ colon (X, Y) \ to (X ', Y')}$ , then for the corresponding homomorphism $f_{*k}\colon H_{k}(X,Y)\to H_{k}(X',Y')$ ${\ displaystyle f _ {* k} \ colon H_ {k} (X, Y) \ to H_ {k} (X ', Y')}$ right $d_{*k}f_{*k}=f_{*k-1}d_{*k}$ ${\ displaystyle d _ {* k} f _ {* k} = f _ {* k-1} d _ {* k}}$ for any dimension $k$ ${\ displaystyle k}$ .
Let be $i\colon Y\to X$ ${\ displaystyle i \ colon Y \ to X}$ and $j\colon X\to (X,Y)$ ${\ displaystyle j \ colon X \ to (X, Y)}$ - attachments, $i_{*k}\colon H_{k}(Y)\to H_{k}(X)$ ${\ displaystyle i _ {* k} \ colon H_ {k} (Y) \ to H_ {k} (X)}$ and $j_{*k}\colon H_{k}(X)\to H_{k}(X,Y)$ ${\ displaystyle j _ {* k} \ colon H_ {k} (X) \ to H_ {k} (X, Y)}$ Are the corresponding homomorphisms, $d_{*k}\colon H_{k}(X,Y)\to H_{k-1}(Y)$ ${\ displaystyle d _ {* k} \ colon H_ {k} (X, Y) \ to H_ {k-1} (Y)}$ Is a boundary homomorphism. Then the sequence determined by them
$\ldots \to H_{k}(Y)\to H_{k}(X)\to H_{k}(X,Y)\to H_{k-1}(Y)\to \ldots$ ${\ displaystyle \ ldots \ to H_ {k} (Y) \ to H_ {k} (X) \ to H_ {k} (X, Y) \ to H_ {k-1} (Y) \ to \ ldots}$
exact ( axiom of accuracy ).
If display $f,g\colon (X,Y)\to (X',Y')$ ${\ displaystyle f, g \ colon (X, Y) \ to (X ', Y')}$ are homotopic , then the corresponding homomorphisms are equal $f_{*k}=g_{*k}$ ${\ displaystyle f _ {* k} = g _ {* k}}$ for any dimension $k$ ${\ displaystyle k}$ ( axiom of homotopy invariance ).
Let be $U\subset X$ ${\ displaystyle U \ subset X}$ - open subset $X$ ${\ displaystyle X}$ , and its closure is contained in the interior of the set $Y$ ${\ displaystyle Y}$ then if the pairs $(X\setminus U,Y\setminus U)$ ${\ displaystyle (X \ setminus U, Y \ setminus U)}$ and $(X,Y)$ ${\ displaystyle (X, Y)}$ belong to an admissible class, then for any dimension $k$ ${\ displaystyle k}$ investment $(X\setminus U,Y\setminus U)\hookrightarrow (X,Y)$ ${\ displaystyle (X \ setminus U, Y \ setminus U) \ hookrightarrow (X, Y)}$ corresponds to an isomorphism $H_{k}(X\setminus U,Y\setminus U)\simeq H_{k}(X,Y)$ ${\ displaystyle H_ {k} (X \ setminus U, Y \ setminus U) \ simeq H_ {k} (X, Y)}$ ( axiom of excision ).
For single point space $H_{k}(*)=0$ ${\ displaystyle H_ {k} (*) = 0}$ for all dimensions $k>0$ ${\ displaystyle k> 0}$ . Abelian group $G=H_{0}(*)$ ${\ displaystyle G = H_ {0} (*)}$ called a group of coefficients ( axiom of dimension ).

For singular homology, an admissible class of pairs consists of all pairs of topological spaces. Previously defined groups of singular homologies with coefficients in the group $G$ ${\ displaystyle G}$ their mappings and boundary homomorphism $d_{*}$ ${\ displaystyle d _ {*}}$ satisfy all these axioms. If we take the class of polyhedra as an admissible class, then we can prove that the homology defined using this system of axioms coincides with simplicial.

Similarly, we can introduce a system of axioms for cohomology, which is completely analogous.

You only need to keep in mind that mapping $f\colon (X,Y)\to (X',Y')$ ${\ displaystyle f \ colon (X, Y) \ to (X ', Y')}$ corresponds to $f^{*k}\colon H^{k}(X',Y')\to H^{k}(X,Y)$ ${\ displaystyle f ^ {* k} \ colon H ^ {k} (X ', Y') \ to H ^ {k} (X, Y)}$ ( contravariance ) and that the boundary homomorphism $\delta ^{*k}\colon H^{k-1}(Y)\to H^{k}(X,Y)$ ${\ displaystyle \ delta ^ {* k} \ colon H ^ {k-1} (Y) \ to H ^ {k} (X, Y)}$ increases dimension.

Extraordinary homologies

In the Steenrod – Eilenberg axiom system, the axiom of dimension is not as important as the others.

Theories of (co) homology that can have nonzero (co) homology groups of a one-point space for dimensions $k>0$ ${\ displaystyle k> 0}$ are called extraordinary or generalized. The most important extraordinary theories are the Atiyah K-theory (note the important contribution to this theory of Hirzebruch , Bott and Adams ) and R. Thom's bordism theory.

Notes

↑ ¹ ² Fomenko, Fuchs, 1989 , p. 95.
↑ Hatcher, 2002 , p. 97.

Literature

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[_8aa3fc956b89b376-1] ¹ ² Fomenko, Fuchs, 1989 , p. 95.

[_d4c46b848f704a3e-2] Hatcher, 2002 , p. 97.