Homotopy group

In mathematics , more precisely in algebraic topology , the homotopy groups are a tool to classify topological spaces . The continuous mappings of an n -dimensional sphere in a given space are grouped into equivalence classes , the so-called homotopy classes . Two images are called homotop if they can be continuously converted into one another. This homotopy classes form a group , the n th homotopy is called the space.

The first homotopy group is also called the fundamental group .

Homotopy-equivalent topological spaces have isomorphic homotopy groups. If two spaces have different homotopy groups, then they cannot be homotopy-equivalent , and therefore not homeomorphic either . For CW-complexes , a partial inversion also applies according to a Whitehead theorem.

definition

In the sphere we choose a point that we call the base point . Be a topological space and a base point. We define it as the set of homotopy classes of continuous maps (i.e. it is ). More precisely, the equivalence classes are defined by homotopias that fix the base point. We could equivalently define the set of images , i.e. H. those continuous mappings of the n -dimensional unit cube that map the edge of the cube into the point . ${\displaystyle S^{n}}$ ${\displaystyle a}$ ${\displaystyle X}$ ${\displaystyle b\in X}$ ${\displaystyle \pi _{n}(X,b)}$ ${\displaystyle f\colon (S^{n},a)\to (X,b)}$ ${\displaystyle f(a)=b}$ ${\displaystyle \pi _{n}(X,b)}$ ${\displaystyle g\colon (I^{n},\partial I^{n})\to (X,b)}$ ${\displaystyle X}$ ${\displaystyle b}$

For the set of homotopy classes can be given a group structure. The construction of the group structure of is similar to that in the case , i.e. the fundamental group . The idea of ​​the construction of the group operation in the fundamental group is to walk through paths one after the other , in the more general -th homotopy group we proceed in a similar way, only that we now glue -cubes together along one side, i.e. H. We define the sum of two pictures by ${\displaystyle n\geq 1}$ ${\displaystyle \pi _{n}(X,b)}$ ${\displaystyle n=1}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle f,g\colon (I^{n},\partial I^{n})\to (X,b)}$

${\displaystyle (f*g)(t)={\begin{cases}f(t_{1},\ldots ,t_{n-1},2t_{n})&t_{n}\leq {\frac {1}{2}}\\g(t_{1},\ldots ,t_{n-1},2t_{n}-1)&t_{n}\geq {\frac {1}{2}}\end{cases}}}$

In the representation by spheres, the sum of two homotopy classes is the homotopy class of the mapping that is obtained when the sphere is first contracted along the equator and then applied to the upper sphere f and the lower g . More precisely: is the composition of the 'lashing the equator together' ( one-point union ) and the figure . ${\displaystyle f+g}$ ${\displaystyle S^{n}\to S^{n}\vee S^{n}}$ ${\displaystyle f\vee g\colon S^{n}\vee S^{n}\to X}$

Is , so is an Abelian group . To prove this fact one should note that two homotopies from dimension two onwards can be "rotated" around one another. For this is not possible, since the edge of not path-connected. ${\displaystyle n\geq 2}$ ${\displaystyle \pi _{n}(X,b)}$ ${\displaystyle n=1}$ ${\displaystyle I^{1}}$

Examples

Homotopy groups of spheres

For true , for it follows from the Hopf theorem that ${\displaystyle 0<k<n}$ ${\displaystyle \pi _{k}(S^{n})=0}$ ${\displaystyle k=n}$

${\displaystyle \pi _{n}(S^{n})=\mathbb {Z} }$

is. Jean-Pierre Serre has proven that must be for a finite group . ${\displaystyle \pi _{k}(S^{n})}$ ${\displaystyle k\not =n,2n-1}$

Eilenberg-MacLane rooms

Topological spaces , the all meet hot Eilenberg-MacLane spaces with . ${\displaystyle X}$ ${\displaystyle \pi _{k}(X)=0}$ ${\displaystyle k\not =0,n}$ ${\displaystyle K(\pi ,n)}$ ${\displaystyle \pi :=\pi _{n}(X)}$

Examples of -spaces are closed, orientable surfaces with the exception of the , closed, orientable, prime 3-manifolds with the exception of and all CAT (0) -spaces , including locally-symmetric spaces of non-compact type , especially hyperbolic manifolds . ${\displaystyle K(\pi ,1)}$ ${\displaystyle S^{2}}$ ${\displaystyle S^{2}\times S^{1}}$

The long exact sequence of a fiber

If there is a Serre fiber with fiber , that is, a continuous image that has the homotopy elevation property for CW complexes , then there is a long, exact sequence of homotopy groups ${\displaystyle p\colon (E,e_{0})\to (B,b_{0})}$ ${\displaystyle F}$

${\displaystyle \ldots \to \pi _{n}(F)\to \pi _{n}(E)\to \pi _{n}(B)\to \pi _{n-1}(F)\to \ldots \to \pi _{0}(E)\to \pi _{0}(B)\to 0}$

The images in question are not group homomorphisms here , since they are not group-valued, but they are exact in the sense that the image resembles the core (the component of the base point is the marked element). ${\displaystyle \pi _{0}}$ ${\displaystyle \pi _{0}}$

Example: the Hopf fiber

The base is here and the total space is . Be the hop figure that the fiber has. From the long exact sequence ${\displaystyle B}$ ${\displaystyle S^{2}}$ ${\displaystyle E}$ ${\displaystyle S^{3}}$ ${\displaystyle p\colon S^{3}\to S^{2}}$ ${\displaystyle S^{1}}$

${\displaystyle \ldots \to \pi _{n}(S^{1})\to \pi _{n}(S^{3})\to \pi _{n}(S^{2})\to \pi _{n-1}(S^{1})\to \ldots }$

and the fact that for , it follows that for holds. In particular is ${\displaystyle \pi _{n}(S^{1})=0}$ ${\displaystyle n\geq 2}$ ${\displaystyle \pi _{n}(S^{2})=\pi _{n}(S^{3})}$ ${\displaystyle n\geq 3}$ ${\displaystyle \pi _{3}(S^{2})=\mathbb {Z} .}$

n equivalences and weak equivalences. Whitehead's Theorem

A continuous mapping is called -equivalence if the induced mapping is for an isomorphism and for a surjection. If the mapping is an isomorphism for all , the mapping is called a weak equivalence. ${\displaystyle f\colon X\to Y}$ ${\displaystyle n}$ ${\displaystyle \pi _{k}(X)\to \pi _{k}(Y)}$ ${\displaystyle k<n}$ ${\displaystyle k=n}$ ${\displaystyle k}$

One theorem from JHC Whitehead says that a weak equivalence between related CW complexes is already a homotopy equivalence . If and dimension is smaller than , then it is sufficient that there is an equivalence. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle n}$

Homotopy and homology. Hurewicz's theorem

For dotted spaces there are canonical homomorphisms from the homotopy groups to the reduced homology groups ${\displaystyle X}$

${\displaystyle h_{n}\colon \pi _{n}(X)\to {\tilde {H}}_{n}(X,\mathbb {Z} ),}$

the Hurewicz homomorphisms (after Witold Hurewicz ) are called. A sentence by Hurewicz says: Is a -contiguous space, i. H. applies for , the Hurewicz homomorphism is in the case of the Abelisierung and an isomorphism. ${\displaystyle X}$ ${\displaystyle (n-1)}$ ${\displaystyle \pi _{k}(X)=0}$ ${\displaystyle k<n}$ ${\displaystyle h_{n}}$ ${\displaystyle n=1}$ ${\displaystyle n>1}$

Relative homotopy groups

One can also define relative homotopy groups for pairs of spaces , their elements are homotopy classes of mappings , two such mappings and are called homotopic if there is a homotopy . The absolute homotopy groups are obtained in a special case . ${\displaystyle \pi _{n}(X,A,a)}$ ${\displaystyle (X,A)}$ ${\displaystyle (B^{n},S^{n-1},b)\to (X,A,a)}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle F\colon (B^{n}\times I,S^{n-1}\times I,b\times I)\to (X,A,a)}$ ${\displaystyle A=\{a\}}$

There is a long exact sequence for each room pair

${\displaystyle \ldots \to \pi _{n+1}(X,A)\to \pi _{n}(A)\to \pi _{n}(X)\to \pi _{n}(X,A)\to \ldots \to \pi _{0}(X)}$

literature

• JP May, A Concise Course in Algebraic Topology . University of Chicago Press, Chicago 1999. ISBN 0-226-51183-9 .

Web links

• Putman: Homotopy groups of spheres and low-dimensional topology

swell

1. ↑ It is important to only allow homotopies here that fix the base point. The set of free homotopy classes has no natural group structure and they are generally not in bijection too . One has a surjective mapping under which two elements correspond to the same free homotopy class if and only if they lie in the same orbit of the action of on . ${\displaystyle \left[S^{n},X\right]}$ ${\displaystyle \pi _{n}(X,b)}$ ${\displaystyle \pi _{n}(X,b)\to \left[S^{n},X\right]}$ ${\displaystyle \pi _{1}(X,b)}$ ${\displaystyle \pi _{n}(X,b)}$
2. ^ JP May, A Concise Course in Algebraic Topology . University of Chicago Press, Chicago 1999. ISBN 0-226-51183-9 , section 9.6
3. ^ JP May, A Concise Course in Algebraic Topology . University of Chicago Press, Chicago 1999. ISBN 0-226-51183-9 , section 10.3
4. ^ JP May, A Concise Course in Algebraic Topology . University of Chicago Press, Chicago 1999. ISBN 0-226-51183-9 , section 15.1