Homeotopy

Short description: Type of homotopy group of a topological space

In algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space.

Definition

The homotopy group functors [math]\displaystyle{ \pi_k }[/math] assign to each path-connected topological space [math]\displaystyle{ X }[/math] the group [math]\displaystyle{ \pi_k(X) }[/math] of homotopy classes of continuous maps [math]\displaystyle{ S^k\to X. }[/math]

Another construction on a space [math]\displaystyle{ X }[/math] is the group of all self-homeomorphisms [math]\displaystyle{ X \to X }[/math], denoted [math]\displaystyle{ {\rm Homeo}(X). }[/math] If X is a locally compact, locally connected Hausdorff space then a fundamental result of R. Arens says that [math]\displaystyle{ {\rm Homeo}(X) }[/math] will in fact be a topological group under the compact-open topology.

Under the above assumptions, the homeotopy groups for [math]\displaystyle{ X }[/math] are defined to be:

[math]\displaystyle{ HME_k(X)=\pi_k({\rm Homeo}(X)). }[/math]

Thus [math]\displaystyle{ HME_0(X)=\pi_0({\rm Homeo}(X))=MCG^*(X) }[/math] is the mapping class group for [math]\displaystyle{ X. }[/math] In other words, the mapping class group is the set of connected components of [math]\displaystyle{ {\rm Homeo}(X) }[/math] as specified by the functor [math]\displaystyle{ \pi_0. }[/math]

Example

According to the Dehn-Nielsen theorem, if [math]\displaystyle{ X }[/math] is a closed surface then [math]\displaystyle{ HME_0(X)={\rm Out}(\pi_1(X)), }[/math] i.e., the zeroth homotopy group of the automorphisms of a space is the same as the outer automorphism group of its fundamental group.

References

• McCarty, G.S. (1963). "Homeotopy groups". Transactions of the American Mathematical Society 106 (2): 293–304. doi:10.1090/S0002-9947-1963-0145531-9. https://www.ams.org/tran/1963-106-02/S0002-9947-1963-0145531-9/S0002-9947-1963-0145531-9.pdf. 
• Arens, R. (1946). "Topologies for homeomorphism groups". American Journal of Mathematics 68 (4): 593–610. doi:10.2307/2371787.