Homotopy

This article covers the topological term homotopy. For the formation of the term homological algebra that resulted from this, see homotopy (homological algebra) . For homotopic groups in chemistry, see topicity .

A homotopy that transforms a coffee cup into a donut (a full torus ).

In topology , a homotopy (from the Greek ὁμός homos 'equal' and τόπος tópos 'place', 'place') is a constant deformation between two images from one topological space to another, for example the deformation of a curve into another curve. One application of homotopy is the definition of homotopy groups , which are important invariants in algebraic topology .

The term “homotopy” describes both the property of two images of being homotopic (preferred) to one another and the image (“continuous deformation”) that conveys this property.

definition

A homotopy between two continuous maps is a continuous map ${\displaystyle f,\,g\colon X\to Y}$

${\displaystyle H\colon X\times {[0,1]}\to Y}$

with the property

${\displaystyle H(x,0)=f(x)}$ and ${\displaystyle H(x,1)=g(x)}$

where is the unit interval . The first parameter corresponds to that of the original images and the second indicates the degree of deformation . The definition becomes particularly clear if one imagines the second parameter as “time” (see picture). ${\displaystyle [0,1]}$

They say that is homotopic to and writes . Homotopy is an equivalence relation on the set of continuous mappings , the associated equivalence classes are called homotopy classes , the set of these equivalence classes is often referred to as. ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f\sim g}$ ${\displaystyle X\to Y}$ ${\displaystyle [X,Y]}$

A continuous mapping is called null homotopic if it is homotopic to a constant mapping . ${\displaystyle f\colon X\to Y}$

example

Homotopy of a circle in R² to a point

Be the unit circle in the plane and the whole plane. Let the mapping be the embedding of in , and be the mapping that maps entirely to the origin ${\displaystyle X=S^{1}\subset \mathbb {R} ^{2}}$ ${\displaystyle Y=\mathbb {R} ^{2}}$ ${\displaystyle f}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle g}$ ${\displaystyle X}$

${\displaystyle f\colon X\to Y}$, And , . ${\displaystyle f(x)=x}$ ${\displaystyle g\colon X\to Y}$ ${\displaystyle g(x)=0}$

Then and are homotopic to one another. Because ${\displaystyle f}$ ${\displaystyle g}$

${\displaystyle H\colon X\times [0,1]\to \mathbb {R} ^{2}}$ With ${\displaystyle H(x,t)=(1-t)\cdot f(x)}$

is steady and fulfilled and . ${\displaystyle H(x,0)=1\cdot f(x)=f(x)}$ ${\displaystyle H(x,1)=0\cdot f(x)=0=g(x)}$

Relative homotopy

Is a subset of , and if two continuous mappings on agree, then and are called homotop relative to , if there is a homotopy for which for each is independent of . ${\displaystyle E}$ ${\displaystyle X}$ ${\displaystyle f,g\colon X\to Y}$ ${\displaystyle E}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle E}$ ${\displaystyle H\colon f\sim g}$ ${\displaystyle H(e,t)}$ ${\displaystyle e\in E}$ ${\displaystyle t}$

Homotopy of two curves

The two dashed paths shown here are homotopic relative to their endpoints. The animation represents a possible homotopy.

An important special case is the homotopy of paths relative to the end points: A path is a continuous mapping ; where is the unit interval. Two paths are called homotopic relative of the endpoints if they are homotopic relative , i.e. H. when the homotopy fixes the starting and ending points. (Otherwise paths in the same path connection component would always be homotopic.) If and are two paths in with and , then a homotopy relative to the endpoints between them is a continuous mapping ${\displaystyle \gamma \colon [0,1]\to X}$ ${\displaystyle [0,1]}$ ${\displaystyle \{0,1\}}$ ${\displaystyle \gamma _{0}}$ ${\displaystyle \gamma _{1}}$ ${\displaystyle Y}$ ${\displaystyle \gamma _{0}(0)=\gamma _{1}(0)=x}$ ${\displaystyle \gamma _{0}(1)=\gamma _{1}(1)=y}$

${\displaystyle H:[0,1]\times [0,1]\to Y}$

with , , and . ${\displaystyle H(t,0)=\gamma _{0}(t)}$ ${\displaystyle H(t,1)=\gamma _{1}(t)}$ ${\displaystyle H(0,s)=x}$ ${\displaystyle H(1,s)=y}$

A path is called null homotop if and only if it is homotop to the constant way . ${\displaystyle \gamma (t)=x_{0}}$

The other common case is the homotopy of mappings between dotted spaces . If and are dotted spaces, then two continuous mappings are homotopic than mappings of dotted spaces if they are relatively homotopic. ${\displaystyle (X,x_{0})}$ ${\displaystyle (Y,y_{0})}$ ${\displaystyle f,g\colon (X,x_{0})\to (Y,y_{0})}$ ${\displaystyle x_{0}}$

Example: the fundamental group

The set of homotopy classes of mappings of dotted spaces from to is the fundamental group from to the base point . ${\displaystyle (S^{1},*)}$ ${\displaystyle (X,x_{0})}$ ${\displaystyle X}$ ${\displaystyle x_{0}}$

For example, if a circle with any selected point , then the path that is described by going around the circle once is not homotopic to the path that is obtained by standing still at the starting point . ${\displaystyle (X,x_{0})}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$

Homotopy equivalence

→ Main article : Homotopy equivalence

Be and two topological spaces and are and continuous mappings. Then the links and are respectively continuous mappings of or on themselves, and one can try to homotopy them to identity on X or Y. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle g\colon Y\to X}$ ${\displaystyle g\circ f}$ ${\displaystyle f\circ g}$ ${\displaystyle X}$ ${\displaystyle Y}$

If there are such and there that homotopic to and to homotopic is, it is called and homotopy equivalent or the same homotopy type . The maps and are then called homotopy equivalences . ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle g\circ f}$ ${\displaystyle \operatorname {id} _{X}}$ ${\displaystyle f\circ g}$ ${\displaystyle \operatorname {id} _{Y}}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle g}$

Homotopy-equivalent spaces have most of the topological properties in common. If and are homotopy equivalent, then applies ${\displaystyle X}$ ${\displaystyle Y}$

• if path-related , so too . ${\displaystyle X}$ ${\displaystyle Y}$
• if and path-connected, the fundamental groups and the higher homotopy groups are isomorphic. ${\displaystyle X}$ ${\displaystyle Y}$
• the homology and cohomology groups of and are the same. ${\displaystyle X}$ ${\displaystyle Y}$
• ${\displaystyle X}$and are deformation retracts of a topological space . ${\displaystyle Y}$ ${\displaystyle Z}$

Isotopy

definition

If two given homotopic mappings and belong to a certain regularity class or have other additional properties, one can ask whether the two can be connected by a path within this class. This leads to the concept of isotopy . An isotopy is a homotopy ${\displaystyle f\colon X\to Y}$ ${\displaystyle g\colon X\to Y}$

${\displaystyle H\colon X\times [0,1]\to Y}$

As above, whereby all intermediate images (for fixed t ) should also have the required additional properties. The associated equivalence classes are called isotope classes . ${\displaystyle H_{t}:=H(\cdot ,t)}$

Examples

So two homeomorphisms are isotopic if a homotopy exists, so that all are homeomorphisms. Two diffeomorphisms are isotopic if they are all diffeomorphisms themselves. (They are then also called diffeotopic .) Two embeddings are isotopic when all embeddings are. ${\displaystyle H_{t}}$ ${\displaystyle H_{t}}$ ${\displaystyle H_{t}}$

Difference from homotopy

Indeed, requiring two mappings to be isotopic may be a stronger requirement than requiring them to be homotopic. For example, the homeomorphism of the unit disk in , which is defined by , is the same as a 180 degree rotation around the origin, therefore the identity map and are isotopic because they can be connected by rotations. In contrast, the mapping on the interval in defined by is not isotopic to identity. This is because each homotopy of the two images has to swap the two endpoints at a certain point in time; at this point they are mapped to the same point and the corresponding map is not a homeomorphism. In contrast, homotop to identity, for example through homotopy , is given by . ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle f(x,y)=(-x,-y)}$ ${\displaystyle f}$ ${\displaystyle \left[-1,1\right]}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f(x)=-x}$ ${\displaystyle f}$ ${\displaystyle H''\colon \left[-1,1\right]\times \left[0,1\right]\to \left[-1,1\right]}$ ${\displaystyle H(x,t)=2tx-x}$

Applications

In geometric topology , isotopies are used to establish equivalence relations.

For example in knot theory - when are two knots and to be considered equal? The intuitive idea of ​​deforming one knot into the other leads to the demand for a path of homeomorphisms: an isotopia that begins with the identity of three-dimensional space and ends with a homeomorphism h , so that h the knot in the knot convicted. Such isotopes of the surrounding space is ambient isotopic or Umgebungsisotopie called. ${\displaystyle K_{1}}$ ${\displaystyle K_{2}}$ ${\displaystyle K_{1}}$ ${\displaystyle K_{2}}$

Another important application is the definition of the mapping class group Mod (M) a manifold M . One considers diffeomorphisms of M “up to isotopia”, that means that Mod (M) is the ( discrete ) group of diffeomorphisms of M , modulo the group of diffeomorphisms that are isotopic to identity.

Homotopy can be used in numerical mathematics for a robust initialization for the solution of differential-algebraic equations (see homotopy method ).

Chain homotopy

Two chain homomorphisms

${\displaystyle f_{\bullet },g_{\bullet }\colon (A_{\bullet },d_{A,\bullet })\to (B_{\bullet },d_{B,\bullet })}$

between chain complexes and are called chain homotop if there is a homomorphism ${\displaystyle (A_{\bullet },d_{A,\bullet })}$ ${\displaystyle (B_{\bullet },d_{B,\bullet })}$

${\displaystyle K_{\bullet }\colon (A_{\bullet })\to (B_{\bullet +1})}$

With

${\displaystyle d_{B,{\bullet +1}}K_{\bullet }+K_{\bullet -1}d_{A,{\bullet }}=f_{\bullet }-g_{\bullet }}$

gives.

If there are homotopic maps between topological spaces, then the induced maps are the singular chain complexes ${\displaystyle f,g\colon X\to Y}$

${\displaystyle f_{\bullet },g_{\bullet }\colon (C_{\bullet }(X),d_{\bullet })\to (C_{\bullet }(Y),d_{\bullet })}$

chain homotop.

Dotted homotopy

→ Main article : Dotted topological space

Two dotted figures

${\displaystyle f,g\colon (X,x_{0})\to (Y,y_{0})}$

hot homotopic if there is a continuous map with ${\displaystyle H\colon X\times \left[0,1\right]\to Y}$

${\displaystyle H(x,0)=f(x)}$and for everyone ${\displaystyle H(x,1)=g(x)}$ ${\displaystyle x\in X}$

${\displaystyle H(x_{0},t)=y_{0}}$ for all ${\displaystyle t\in \left[0,1\right]}$

gives. The set of homotopy classes of dotted maps is denoted by. ${\displaystyle \left[X,Y\right]}$

literature

• Brayton Gray: Homotopy theory. An introduction to algebraic topology (=  Pure and Applied Mathematics . No. 64 ). Academic Press, New York et al. a. 1975, ISBN 0-12-296050-5 .
• Allen Hatcher: Algebraic Topology . Cambridge University Press, Cambridge 2002, ISBN 0-521-79540-0 .
• John McCleary (Ed.): Higher Homotopy Structures in Topology and Mathematical Physics . Proceedings of an international Conference, June 13-15, 1996 at Vassar College, Poughkeepsie, New York, to Honor the sixtieth Birthday of Jim Stasheff. American Mathematical Society, Providence RI 1999, ISBN 0-8218-0913-X ( Contemporary Mathematics 227).
• George W. Whitehead: Elements of Homotopy Theory . Corrected 3rd printing. Springer, New York a. a. 1995, ISBN 0-387-90336-4 ( Graduate Texts in Mathematics 61).
• M. Sielemann, F. Casella, M. Otter, C. Claus, J. Eborn, SE Mattsson, H. Olsson: Robust Initialization of Differential-Algebraic Equations Using Homotopy . International Modelica Conference, Dresden 2011, ISBN 978-91-7393-096-3 .

Individual evidence

1. ↑ Tammo tom Dieck : Topology . 2nd Edition. de Gruyter, Berlin 2000, ISBN 3-11-016236-9 , p. 277 .