Axiomatic homology

The term homology theory comes from algebraic topology and axiomatically characterizes the way in which singular homology or the Bordism theories assign Abelian groups to topological spaces (homology groups, see homology theory ). The term axiomatic homology summarizes the investigation of those homology theories that satisfy the Eilenberg-Steenrod axioms .

Eilenberg-Steenrod axioms

Functors and natural transformations

Let for all functors be from the category of topological pairs of spaces (ie pairs of topological spaces such that ) to the category of Abelian groups. For an illustration , the abbreviation is denoted by. A mapping from a pair of rooms to a pair of rooms is a continuous mapping from to , so that . Furthermore, let us define a natural transformation from the functor to the functor for each , with that functor belonging to the category of space pairs in itself that assigns the space pair to each space pair . So assign a homomorphism to each pair of spaces . Here and in the following denotes the pair of rooms . Written out, these conditions form the first three Eilenberg-Steenrod axioms: ${\displaystyle H_{n}}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle (X,A)}$ ${\displaystyle A\subset X}$ ${\displaystyle f}$ ${\displaystyle H_{\,n\,}(f)}$ ${\displaystyle f_{*}}$ ${\displaystyle f}$ ${\displaystyle (X,A)}$ ${\displaystyle (Y,B)}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f(A)\subset B}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle \partial _{n}}$ ${\displaystyle H_{n}}$ ${\displaystyle H_{n-1}\circ P}$ ${\displaystyle P}$ ${\displaystyle (X,A)}$ ${\displaystyle (A,\varnothing )}$ ${\displaystyle (X,A)}$ ${\displaystyle \partial _{n}}$ ${\displaystyle H_{n}(X,A)\rightarrow H_{n-1}(A)}$ ${\displaystyle A}$ ${\displaystyle (A,\varnothing )}$

1) If is equal to identity, then is also equal to identity ${\displaystyle f\colon (X,A)\rightarrow (X,A)}$ ${\displaystyle f_{*}\colon H_{n}(X,A)\rightarrow H_{n}(X,A)}$

2) For two figures and applies ${\displaystyle f\colon (X,A)\rightarrow (Y,B)}$ ${\displaystyle g\colon (Y,B)\rightarrow (Z,C)}$ ${\displaystyle (g\circ f)_{*}=g_{*}\circ f_{*}}$

3) ${\displaystyle \partial _{n}\circ f_{*}=(f|A)_{*}\circ \partial _{n}}$

Further axioms

The more substantive-topological axioms that were designed directly on the model of singular and simplicial homology are the following three:

4) Axiom of Exactness: There is a long exact sequence of groups:

${\displaystyle \cdots \rightarrow H_{n}(A)\rightarrow H_{n}(X)\rightarrow H_{n}(X,A)\rightarrow H_{n-1}(A)\rightarrow H_{n-1}(X)\rightarrow H_{n-1}(X,A)\cdots ,}$

The images and are each induced by the corresponding inclusions. The mapping is defined by the natural transformation . ${\displaystyle H_{n}(A)\rightarrow H_{n}(X)}$ ${\displaystyle H_{n}(X)\rightarrow H_{n}(X,A)}$ ${\displaystyle H_{n}(X,A)\rightarrow H_{n-1}(A)}$ ${\displaystyle \partial _{n}}$

5) Homotopy axiom: Let two continuous mappings be homotopy . Then the two induced group homomorphisms are identical. ${\displaystyle f,g\colon (X,A)\rightarrow (Y,B)}$ ${\displaystyle f_{*},g_{*}\colon H_{n}(X,A)\rightarrow H_{n}(Y,B)}$

6) Ausschneidungsaxiom: Be a pair of spaces and so that the completion of is contained in the interior of . Then the mapping induced by the inclusion is an isomorphism . ${\displaystyle (X,A)}$ ${\displaystyle B\subset A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle H_{n}(X-B,A-B)\rightarrow H_{n}(X,A)}$

A family of functors and natural transformations that fulfill the above axioms is called homology theory or generalized homology theory . If you turn all arrows around in the axioms, i.e. if you consider contravariant functors , you get the axioms for a cohomology theory . ${\displaystyle H^{n}}$

Dimensional axiom

Classically, the so-called dimensional axiom was added to the axioms mentioned:

7) It applies

${\displaystyle H_{m}(pt)={\begin{cases}G&m=0\\0&{\mbox{sonst}}\end{cases}}}$

for an Abelian group . ${\displaystyle G}$

Only then was a family of functors and natural transformations called a homology theory. This is what happened in the book Foundations of Algebraic Topology by Eilenberg and Steenrod from 1952, where these axioms were first discussed. At that time, only homology theories that fulfilled the dimensional axiom were known. However, other examples were discovered later, as explained under Examples. In general, the homology groups of a point are called the coefficients of a homology theory.

Inferences

Simple conclusions

Direct implications are that for all and to Ausschneidungssatz and for homotopy equivalent to . From this it follows also for homotopy equivalent to . ${\displaystyle H_{n}(X,X)=0}$ ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle H_{n}(X,A)\cong H_{n}(Y,B)}$ ${\displaystyle (X,A)}$ ${\displaystyle (Y,B)}$ ${\displaystyle H_{n}(X,A)=0}$ ${\displaystyle A}$ ${\displaystyle X}$

Mayer-Vietoris sequence

A very practical aid is the so-called Mayer-Vietoris sequence, which can be proven by chasing a diagram from the axiom of cutout and exactness. This means that for a space , two closed subsets and , so that the union of the interior of with the interior of is equal , and a subset, the following sequence is exact : ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle X}$ ${\displaystyle C\subset A\cap B}$

${\displaystyle \cdots \rightarrow H_{n}(A\cap B,C)\rightarrow H_{n}(A,C)\oplus H_{n}(B,C)\rightarrow H_{n}(X,C)\rightarrow H_{n-1}(A\cap B,C)}$

${\displaystyle \rightarrow H_{n-1}(A,C)\oplus H_{n-1}(B,C)\cdots \,}$

A simple application is that what you just in the sequence and the two copies of with or referred to (the intersection is empty, so also ). ${\displaystyle H_{n}(Y\coprod Y)\cong H_{n}(Y)\oplus H_{n}(Y)}$ ${\displaystyle X=Y\coprod Y}$ ${\displaystyle Y}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$

Note: The Mayer-Vietoris sequence applies to homology theories if the inclusions induce isomorphisms on the homology groups of . This is especially the case with the above requirement because of the cutting axiom. ${\displaystyle H}$ ${\displaystyle (A,A\cap B)\rightarrow (X,B),(B,A\cap B)\rightarrow (X,A)}$ ${\displaystyle H}$

Suspension isomorphism

Choice of A, B and C.

With the help of the Mayer-Vietoris sequence one can also prove that the hanging isomorphism is valid, where the hanging of denotes and a point in . To do this, one sets in the Mayer-Vietoris sequence and as in the drawing and a point in the intersection of and . The subspaces and are both homotopy equivalent to one point, their intersection too . The exact sequence becomes like this: ${\displaystyle H_{n+1}(SY,pt)\cong H_{n}(Y,pt)}$ ${\displaystyle SY}$ ${\displaystyle Y}$ ${\displaystyle pt}$ ${\displaystyle Y}$ ${\displaystyle X=SY,A}$ ${\displaystyle B}$ ${\displaystyle C}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A\cap B}$ ${\displaystyle Y}$

${\displaystyle \cdots \rightarrow H_{n}(Y,pt)\rightarrow 0\rightarrow H_{n}(SY,pt)\rightarrow H_{n-1}(Y,pt)\rightarrow 0\cdots \,}$

This shows the required isomorphism.

Homology of the spheres

If one now assumes that the dimensional axiom also applies, one can calculate the homology of the sphere with it . It only consists of two points. It therefore applies according to the cut-out theorem and for n> 0. According to the suspension isomorphism, it holds inductively for now and for , since the suspension of the (n-1) sphere is the n-sphere. If you now look at the exact sequence for the couple , you get that for for and and otherwise for . For one gets directly what is the same for and otherwise. One can show that one can now calculate the homology of every finite CW complex with the help of the cellular homology . So for finite CW complexes one gets the same results for homology theories that satisfy the dimensional axiom as for singular homology . ${\displaystyle S^{0}}$ ${\displaystyle H_{0}(S^{0},pt)=H_{0}(pt)=G}$ ${\displaystyle H_{n}(S^{0},pt)=0}$ ${\displaystyle H_{n}(S^{n},pt)=G}$ ${\displaystyle H_{i}(S^{n},pt)=0}$ ${\displaystyle i\neq n}$ ${\displaystyle (S^{n},pt)}$ ${\displaystyle H_{i}(S^{n})=G}$ ${\displaystyle i=0}$ ${\displaystyle i=n}$ ${\displaystyle 0}$ ${\displaystyle n>0}$ ${\displaystyle n=0}$ ${\displaystyle H_{i}(S^{0})=H_{i}(pt)\oplus H_{i}(pt)}$ ${\displaystyle G\oplus G}$ ${\displaystyle i=0}$ ${\displaystyle 0}$

Eilenberg-Steenrod uniqueness theorem

The historical situation in 1945, when Eilenberg and Steenrod first published the Eilenberg-Steenrod axioms mentioned above, was that there were several proposals for defining the homology of a room, all of which had similar properties and which at least applied to most rooms calculated the same groups. The most prominent example is certainly the singular homology. Further examples are the now almost forgotten Vietoris homology and, on the cohomology side, the Čech cohomology . Eilenberg and Steenrod wanted to put these theories on a common basis and show that they calculate the same groups on a large class of spaces.

In order to formulate their uniqueness theorem precisely, we must first define a natural transformation between two theories of homology. This is a natural transformation between two functors and , both of which form a homology theory that is compatible with the connection homomorphism. That means that for each room pair and each one must apply that the diagram ${\displaystyle T}$ ${\displaystyle h}$ ${\displaystyle H}$ ${\displaystyle (X,A)}$ ${\displaystyle n}$

${\displaystyle {\begin{array}{ccc}H_{n}(X,A)&{\xrightarrow {\ T\ }}&h_{n}(X,A)\\\downarrow \scriptstyle \partial &&\downarrow \scriptstyle \partial \\H_{n-1}(A)&{\xrightarrow {\ T\ }}&h_{n-1}(A)\end{array}}}$

commutes.

Eilenberg and Steenrod's theorem of uniqueness now states that every natural transformation of two homology theories that is an isomorphism on all spheres is also an isomorphism on all finite CW complexes.

This theorem can be accepted under the additional assumption that the two theories of homology use the so-called Milnor or Wedge axiom

${\displaystyle H_{n}(\bigvee _{i\in I}X_{i},pt)=\bigoplus _{i\in I}H_{n}(X_{i},pt)}$

meet, tighten. Then it is true that under the same conditions the natural transformation is an isomorphism on all CW complexes. If one additionally demands that images which induce isomorphisms on all homotopy groups also induce isomorphisms on all homology groups , the natural transformation is even an isomorphism on all topological spaces. ${\displaystyle \pi _{n}(X)}$ ${\displaystyle H_{n}(X,pt)}$

Reduced theories of homology

It turns out that it is useful for many purposes to include the base point in a homology theory without generally defining relative groups. This is particularly useful when comparing homology groups to homotopy groups . These reduced homology theories can be described axiomatically as follows.

Let for each functor be from the category of topological spaces to the category of Abelian groups. Furthermore, there are natural transformations , whereby the insertion functor is on the category of dotted topological spaces. The following axioms should apply: ${\displaystyle {\tilde {H}}_{n}}$ ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle \sigma _{n}\colon {\tilde {H}}_{n}\rightarrow {\tilde {H}}_{n+1}\circ \Sigma }$ ${\displaystyle \Sigma }$

1) Each dotted figure induces a long exact sequence ${\displaystyle f\colon X\rightarrow Y}$

${\displaystyle \cdots \rightarrow {\tilde {H}}_{n}(X)\rightarrow {\tilde {H}}_{n}(Y)\rightarrow {\tilde {H}}_{n}(Cf)\rightarrow {\tilde {H}}_{n}(\Sigma X)\rightarrow {\tilde {H}}_{n}(\Sigma Y)\rightarrow {\tilde {H}}_{n}(\Sigma Cf)\cdots }$

Here is the image cone of f. ${\displaystyle Cf}$

2) If two mappings are homotopic, then applies . ${\displaystyle f,g\colon X\rightarrow Y}$ ${\displaystyle {\tilde {H}}_{n}(f)={\tilde {H}}_{n}(g)}$

3) The natural transformation is an isomorphism for all n and X. ${\displaystyle \sigma _{n}\colon {\tilde {H}}_{n}(X)\rightarrow {\tilde {H}}_{n+1}(\Sigma X)}$

It can be shown that any homology theory by defining a reduced homology theory. The other way round defines a reduced homology theory by means of a homology theory, wherein the inclusion designated. ${\displaystyle H_{n}}$ ${\displaystyle {\tilde {H}}_{n}(X)=H_{n}(X,pt)}$ ${\displaystyle {\tilde {H}}_{n}}$ ${\displaystyle H_{n}(X,A)={\tilde {H}}_{n}(Ci)}$ ${\displaystyle i\colon A\rightarrow X}$

Since the reduced homology of a point is equal to zero in the case of singular homology, the homology of the 0-sphere is referred to here as the coefficients. ${\displaystyle {\tilde {H}}_{n}(S^{0})}$

Examples

Singular homology

→ Main article : Singular homology

The most basic and most important example of a homology theory is the singular homology with coefficients in a group G. It was the first known homology theory that is defined on all topological spaces. As stated in the corresponding article, it satisfies all of the Eilenberg-Steenrod axioms, including the dimensional axiom. The singular homology also fulfills the Milnor axiom and the condition that isomorphisms on homotopy groups induce isomorphisms on homology groups.

Bordism theories

The simplest theory of Bordism is that of unoriented Bordism. It was developed by René Thom in the mid-1950s .

Two compact, unbounded manifolds M and N are called bordant if there is a bounded manifold W such that . One can show that this relation is an equivalence relation . The equivalence classes are called Bordism classes . Using the disjoint union and the Cartesian product , one can define addition and multiplication on the Bordism classes. They thus form a ring. An example for two bordant manifolds is the n- sphere and the empty set, which are bordant by means of the (n + 1) -dimensional full sphere. Examples of manifolds that are not bordant to the empty set are the point and the 2-dimensional real projective space . ${\displaystyle \partial W\cong M\amalg N}$ ${\displaystyle \mathbb {R} P^{2}}$

A " singular p-manifold M in a topological space X" is a pair (M, f), where f is a mapping from M to X and M is a p-dimensional manifold. Two such singular manifolds (M, f) and (N, g) are called bordant if they are bordant over a manifold W and there is a map F from W to X which, restricted to M and N, results in mappings f and g, respectively. The Abelian group generated by the singular p-manifolds, from which the Bordism relation is divided, is denoted by . You can also define relative groups in a similar way. These form a homology theory. The coefficients, ie the homology of one point, are exactly the abovementioned Bordism ring. It shows that it is not zero and that the unoriented Bordism does not fulfill the dimensional axiom. ${\displaystyle MO_{p}(X)}$ ${\displaystyle MO_{p}(X,A)}$ ${\displaystyle \mathbb {R} P^{2}}$ ${\displaystyle MO_{2}(pt)}$

If you provide the manifolds with additional structures, such as an orientation or an almost complex structure , you get many more examples of Bordism theories.

Stable homotopy theory

The homotopy groups of a room do not form a reduced homology theory. They obviously satisfy the homotopy axiom, but Freudenthal's suspension theorem guarantees the suspension isomorphism only in a certain area. The long exact sequence also creates difficulties. ${\displaystyle \pi _{n}(X)}$

Using Freudenthal's suspension theorem , however, one can use the homotopy groups to get a reduced theory of homology. According to the suspension theorem , homomorphisms are obtained which, for k> N, are isomorphisms for a suitable N. Here refers to the k-th device to attach . The stable homotopy groups are defined . The suspension isomorphism is now valid by definition and the existence of a long exact sequence can also be shown. ${\displaystyle \pi _{n+k}(\Sigma ^{k}X)\rightarrow \pi _{n+k+1}(\Sigma ^{k+1}X)}$ ${\displaystyle \Sigma ^{k}}$ ${\displaystyle \pi _{n}^{stab}=\pi _{n+N+1}(\Sigma ^{N+1}X)}$

The coefficients of the stable homotopy theory are the stable homotopy groups of the sphere , since the k-th suspension of the 0-sphere results in the k-sphere. These are extremely difficult to calculate and only partially known, although great efforts have been made in this direction.

Spectra

→ Main article : spectrum (topology)

A spectrum is a sequence of dotted spaces with images . Alternatively, you can also specify the adjoint mappings . Here stands for the loop space from , ie the dotted images from the to provided with the compact-open topology . If there is a homotopy equivalence for every n, it is called an omega spectrum. ${\displaystyle {\underline {E}}}$ ${\displaystyle E_{n}}$ ${\displaystyle e_{n}\colon \Sigma E_{n}\rightarrow E_{n+1}}$ ${\displaystyle e_{n}'\colon E_{n}\rightarrow \Omega E_{n+1}}$ ${\displaystyle \Omega E_{n+1}}$ ${\displaystyle E_{n+1}}$ ${\displaystyle S^{1}}$ ${\displaystyle E_{n+1}}$ ${\displaystyle e_{n}'}$ ${\displaystyle {\underline {E}}}$

There is a very close connection between spectra and homology and cohomology theories. One defines

${\displaystyle {\tilde {H}}_{n}^{E}(X)=\lim _{\rightarrow }[\pi _{n}(E_{0}\wedge X)\rightarrow \pi _{n+1}(\Sigma E_{0}\wedge X)\rightarrow \pi _{n+1}(E_{1}\wedge X)\rightarrow \cdots ],}$

so one can show that this forms a reduced homology theory. That stands for the direct Limes and that for the Smash product . On the other hand, each reduced homology theory can in this way by a spectrum represent . ${\displaystyle {\tilde {H}}_{n}^{E}}$ ${\displaystyle \lim _{\rightarrow }}$ ${\displaystyle \wedge }$

For an omega spectrum is a reduced cohomology theory. According to Brown's representation theorem, every reduced cohomology theory can be represented in this way. ${\displaystyle {\underline {E}}}$ ${\displaystyle {\tilde {H}}_{E}^{n}(X)=[X,E_{n}]}$

The representing spectrum for both the singular homology and the singular cohomology with coefficient group G consists of the Eilenberg-MacLane spaces . These are CW complexes which have G as the nth homotopy group and whose other homotopy groups all disappear. Since there is always a , one can always find a homotopy equivalence , which makes it an omega spectrum. ${\displaystyle {\underline {H}}}$ ${\displaystyle K(G,n)}$ ${\displaystyle \Omega K(G,n)}$ ${\displaystyle K(G,n-1)}$ ${\displaystyle \Omega K(G,n)\rightarrow K(G,n-1)}$ ${\displaystyle {\underline {H}}}$

literature

• Samuel Eilenberg & Norman Steenrod : Foundations of Algebraic Topology. Princeton University Press, 1964 (first textbook with the Eilenberg-Steenrod axioms)
• Allen Hatcher: Algebraic Topology . Cambridge University Press, 2002, ISBN 0521795400 (general introduction to algebraic topology)
• Robert M. Switzer: Algebraic Topology - Homology and Homotopy Springer, 2000, ISBN 3540427503 (goes into detail on the theory of the various generalized homology and cohomology theories and that of the spectra)

Web links

• Vietoris homology