Singular homology

The singular homology is a method of algebraic topology that any topological space a sequence of abelian groups assigns. To put it clearly, it counts the different-dimensional holes in a room. Compared to the homotopy groups of a similar type , the singular homology has the advantage that it is much easier to calculate and thus represents the most effective algebraic invariant for many applications . It is defined as the homology to the singular chain complex .

Simplicial homology

The historical roots of singular homology lie in simplicial homology. Let us be a simplicial complex , that is, a set of simplices , so that every 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). ${\ displaystyle X}$

The goal is now to turn this simplicial complex into a chain complex , from which one then takes the homology . Let the free Abelian group be over the set of -simplices of the simplicial complex . The edge mapping in SK maps each simplex to the alternating sum of its side surfaces, that is ${\ displaystyle C_ {n}}$ ${\ displaystyle n}$${\ displaystyle d \ colon C_ {n} \ rightarrow C_ {n-1}}$

${\ displaystyle d ([v_ {0}, \ dots, v_ {n}]) = + [v_ {1}, \ dots, v_ {n}] - [v_ {0}, v_ {2}, \ dots , v_ {n}] + \ cdots \ pm [v_ {0}, \ dots, v_ {n-1}] \ ,,}$

whereby the alternating sign factors can also be interpreted as "geometric orientation variables".

The homology of this chain complex is then called the simplicial homology of${\ displaystyle X}$ .

historical overview

The definition of simplicial homology has two major problems. One is that not every topological space is represented as a simplicial complex. The second and more important is that the same space can have two different representations as a simplicial complex and thus a priori simplicial homology does not represent a topological invariant of the space. Historically, the first attempt at a solution to this problem was the so-called main conjecture that Steinitz and Tietze put forward at the beginning of the 20th century. This says that two triangulations of a space always have a common refinement. However, the main conjecture was refuted by Milnor in 1961 .

However, the solution to the problem took shape in the thirties and forties through the work of Lefschetz and Eilenberg . They defined the singular homology. This is basically similar to simplicial homology, but uses the so-called singular chain complex as its chain complex.

definition

Singular chain complex

Let be a topological space and a - (Euclidean) simplex , a singular -Simplex in is a continuous mapping . With the free Abelian group , which is generated by the set of all singular -Simplices in . An element of is a formal linear combination of singular simplices and is called a singular chain. The group is called the singular chain group of the dimension . ${\ displaystyle X}$${\ displaystyle \ Delta _ {p}}$${\ displaystyle p}$${\ displaystyle p}$${\ displaystyle X}$ ${\ displaystyle a \ colon \ Delta _ {p} \ to X}$${\ displaystyle C_ {p} (X)}$${\ displaystyle p}$${\ displaystyle X}$${\ displaystyle C_ {p} (X)}$${\ displaystyle p}$${\ displaystyle C_ {p} (X)}$${\ displaystyle p}$

For one is through ${\ displaystyle \ sigma \ in C_ {p} (X)}$

${\ displaystyle \ partial _ {p} (\ sigma) = + \ sigma | [v_ {1}, \ dots, v_ {p}] - \ sigma | [v_ {0}, v_ {2}, \ dots, v_ {p}] + \ cdots \ pm \ sigma | [v_ {0}, \ dots, v_ {p-1}]}$

defines a homomorphism . This gives an edge operator , that is, it holds . So is ${\ displaystyle \ partial _ {p} \ colon C_ {p} (X) \ to C_ {p-1} (X)}$${\ displaystyle \ partial _ {p} \ partial _ {p + 1} = 0}$

${\ displaystyle \ ldots {\ stackrel {\ partial _ {3}} {\ longrightarrow}} C_ {2} (X) {\ stackrel {\ partial _ {2}} {\ longrightarrow}} C_ {1} (X ) {\ stackrel {\ partial _ {1}} {\ longrightarrow}} C_ {0} (X)}$

a chain complex called the singular chain complex.

Singular homology

The homology of this chain complex is called the singular homology of${\ displaystyle X}$ or simply the homology of and is called the homology groups ${\ displaystyle X}$

${\ displaystyle H_ {p} (X): = \ ker (\ partial _ {p}) / \ mathop {\ rm {im}} (\ partial _ {p + 1})}$

also more precise than the singular homology groups. For every simplicial complex it is isomorphic to the simplicial homology.

The elements of are referred to as homology classes. ${\ displaystyle H _ {*} (X)}$

Reduced homology

The 0th homologue plays a special role in many theorems of homology theory , which is why it is often useful to consider the reduced homology for a uniform formulation of propositions and proofs . This is defined by ${\ displaystyle H_ {0} (X)}$ ${\ displaystyle {\ tilde {H}} _ {*} (X)}$

${\ displaystyle {\ tilde {H}} _ {i} (X) = H_ {i} (X)}$ for all ${\ displaystyle i \ geq 1}$

and

${\ displaystyle {\ tilde {H}} _ {0} (X) = \ ker (\ epsilon) / \ mathop {\ rm {im}} (\ partial _ {1})}$,

being the through ${\ displaystyle \ epsilon: C_ {0} (X) \ rightarrow \ mathbb {Z}}$

${\ displaystyle \ epsilon \ left (\ sum _ {i = 1} ^ {r} n_ {i} x_ {i} \ right) = \ sum _ {i = 1} ^ {r} n_ {i}}$

defined augmentation of the chain complex . It applies ${\ displaystyle (C _ {*} (X), \ partial _ {*})}$

${\ displaystyle H_ {0} (X) \ simeq {\ tilde {H}} _ {0} (X) \ oplus \ mathbb {Z}}$.

Relative homology and illustrations

The singular homology can be found not only for a space , but also for a pair of spaces , i.e. H. of a space and a space contained in it . To do this, the chain complex is set equal to the factor group , the definition of the edge mapping remains. The homology of this chain complex is called the relative homology group . To put it clearly, one wants to ignore the interior of , as it will be specified in the next section in the cut-out property. It applies . ${\ displaystyle X}$${\ displaystyle (X, A)}$${\ displaystyle X}$${\ displaystyle A \ subset X}$${\ displaystyle C_ {n} (X, A)}$${\ displaystyle C_ {n} (X) / C_ {n} (A)}$${\ displaystyle d}$${\ displaystyle H_ {n} (X, A)}$${\ displaystyle A}$${\ displaystyle H_ {n} (X, \ varnothing) = H_ {n} (X)}$

Each mapping between two pairs of spaces also induces a group homomorphism of the corresponding homology groups. Let us be a continuous mapping between two pairs of spaces, i.e. H. a continuous mapping from to such that . This mapping defines a chain mapping from to by assigning the singular simplex to each singular simplex . This gives you an illustration . So we get that each is a covariant functor from the category of space pairs to the category of Abelian groups. ${\ displaystyle f \ colon (X, A) \ rightarrow (Y, B)}$${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle f (A) \ subset B}$${\ displaystyle f}$${\ displaystyle C_ {n} (X, A)}$${\ displaystyle C_ {n} (Y, B)}$${\ displaystyle \ sigma: \ Delta ^ {n} \ rightarrow X}$${\ displaystyle f \ circ \ sigma}$${\ displaystyle f _ {*} \ colon H_ {n} (X, A) \ rightarrow H_ {n} (Y, B)}$${\ displaystyle H_ {n}}$

properties

With the means of homological algebra one can show that there always exists a long exact sequence of homology 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 induced by inclusion or projection. The mapping is a boundary operator defined by the snake lemma . ${\ 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}}$

Another important property of is its homotopy invariance . Let there be two continuous maps that are homotopic . Then the so-called homotopy law says: The induced group homomorphisms are identical. In particular, the homology groups of two homotopy-equivalent spaces are isomorphic. ${\ displaystyle H_ {n}}$${\ displaystyle f, g \ colon (X, A) \ rightarrow (Y, B)}$${\ displaystyle f _ {*}, g _ {*} \ colon H_ {n} (X, A) \ rightarrow H_ {n} (Y, B)}$

The clipping property applies to relative homology groups. Be this a couple and space , so that the completion of the interior of is included. Then the mapping induced by the inclusion is an isomorphism . ${\ displaystyle (X, A)}$${\ displaystyle B \ subset A}$${\ displaystyle B}$${\ displaystyle A}$${\ displaystyle H_ {n} (XB, AB) \ rightarrow H_ {n} (X, A)}$

The so-called Eilenberg-Steenrod axioms are thus fulfilled and it is shown that the singular homology is a homology theory . This means that all properties that apply in general to all theories of homology also apply to the singular homology. These are in particular the Mayer-Vietoris sequence and the mounting isomorphism, which says that . In this context, the hanging of . ${\ displaystyle H_ {n + 1} (\ Sigma X, pt) \ cong H_ {n} (X, pt)}$${\ displaystyle \ Sigma X}$${\ displaystyle X}$

For an n-dimensional manifold it holds that for . More generally, this also applies to a CW complex that does not have any cells larger than . ${\ displaystyle M}$${\ displaystyle H_ {m} (M) = 0}$${\ displaystyle m> n}$${\ displaystyle n}$

Examples and calculation

The simplest example is the homology of a point. There is only one mapping into space for each simplex , with which the chain complex takes the following shape: ${\ displaystyle \ Delta ^ {n}}$

${\ displaystyle \ cdots \ rightarrow \ mathbb {Z} \ rightarrow \ mathbb {Z} \ rightarrow \ mathbb {Z} \ rightarrow 0.}$

The edge mapping is always alternating between 0 and identity, so that the penultimate arrow is the zero mapping . It therefore holds for every n> 0 and . Because of the homotopy invariance, the same applies to every contractible space. ${\ displaystyle H_ {n} (pt) = 0}$${\ displaystyle H_ {0} (pt) = \ mathbb {Z}}$

In general, however, a direct consideration of the singular chain complex is of little use, since it is normally infinite-dimensional in every positive dimension. One method of calculation is based on the above-mentioned properties of singular homology. For example, with the help of the suspension isomorphism and the long exact sequence of the space pair, one can calculate that for for or , for and otherwise. ${\ displaystyle (S ^ {n}, pt)}$${\ displaystyle n \ neq 0}$ ${\ displaystyle H_ {m} (S ^ {n}) = \ mathbb {Z}}$${\ displaystyle m = 0}$${\ displaystyle m = n}$${\ displaystyle H_ {m} (S ^ {0}) = \ mathbb {Z} ^ {2}}$${\ displaystyle m = 0}$${\ displaystyle H_ {m} (S ^ {n}) = 0}$

Another example that can be calculated using methods of cellular homology is the homology of real projective space . For straight: ${\ displaystyle n}$

${\ displaystyle H_ {m} (\ mathbb {RP} ^ {n}) = {\ begin {cases} \ mathbb {Z} & m = 0 \\\ mathbb {Z} / 2 \ mathbb {Z} & m \ equiv 1 {\ text {(mod 2)}} {\ text {and}} n> m> 0 \\ 0 & m \ equiv 0 {\ text {(mod 2)}} {\ text {and}} n \ geq m > 0 {\ text {or}} m> n \ end {cases}}}$

And for odd: ${\ displaystyle n}$

${\ displaystyle H_ {m} (\ mathbb {RP} ^ {n}) = {\ begin {cases} \ mathbb {Z} & m = 0 {\ text {or}} n \\\ mathbb {Z} / 2 \ mathbb {Z} & m \ equiv 1 {\ text {(mod 2)}} {\ text {and}} n> m> 0 \\ 0 & m \ equiv 0 {\ text {(mod 2)}} {\ text {and}} n> m> 0 {\ text {or}} m> n \ end {cases}}}$

Applications

A classic application is Brouwer's Fixed Point Theorem . This means that every continuous mapping of the n -dimensional sphere D n has a fixed point in itself. The proof runs by contradiction.

Illustration of F in D 2

Suppose there is a map that has no fixed point. Then you can draw the ray from to for each point , which hits the edge of the sphere in the point (as indicated in the picture). The function is continuous and has the property that every point on the edge is mapped onto itself. So is ${\ displaystyle f \ colon D ^ {n} \ rightarrow D ^ {n}}$${\ displaystyle x \ in D ^ {n}}$${\ displaystyle f (x)}$${\ displaystyle x}$${\ displaystyle F (x)}$${\ displaystyle F \ colon D ^ {n} \ rightarrow S ^ {n-1}}$

${\ displaystyle F \ circ \ iota \ colon S ^ {n-1} \ rightarrow D ^ {n} \ rightarrow S ^ {n-1}}$

equal to the identity, whereby the inclusion of the edge in the full sphere is. This is also the induced mapping ${\ displaystyle \ iota}$

${\ displaystyle (F \ circ \ iota) _ {*} \ colon H_ {n-1} (S ^ {n-1}) \ rightarrow H_ {n-1} (D ^ {n}) \ rightarrow H_ { n-1} (S ^ {n-1})}$

equal to identity. Now, however, according to the previous section , however . So we have the contradiction. ${\ displaystyle H_ {n-1} (S ^ {n-1}) \ cong \ mathbb {Z}}$${\ displaystyle H_ {n-1} (D ^ {n}) \ cong 0}$

Further applications are the Borsuk-Ulam theorem and the Jordan-Brouwer decomposition theorem , a generalization of the Jordan curve theorem .

Coefficients and Betti numbers

In the design of the singular chain complex the free Abelian group became so free - module , formed over all singular simplices. The resulting homology is also known as homology with coefficients in . However, it is also possible to choose any other group of Abelian coefficients . This is achieved by the chain complex with tensoriert . The resulting homology is called the homology of the space pair with coefficients in . ${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle G}$${\ displaystyle C (X, A)}$${\ displaystyle G}$ ${\ displaystyle H (X, A; G)}$${\ displaystyle (X, A)}$${\ displaystyle G}$

The conversion of homology with different groups of coefficients into one another is usually carried out by means of universal coefficient theorems .

Bodies play a special role as coefficients. Here the chain complex is a vector space in every dimension and thus also the resulting homology. In this way you can also define the so-called Betti numbers :

${\ displaystyle b_ {i} (X) = \ dim _ {\ mathbb {Q}} H_ {i} (X; \ mathbb {Q})}$