Topological rigidity

In the mathematical field of topology, a manifold M is called topologically rigid if every manifold homotopically equivalent to M is also homeomorphic to M.^[1]

Motivation

A central problem in topology is determining when two spaces are the same i.e. homeomorphic or diffeomorphic. Constructing a morphism explicitly is almost always impractical. If we put further condition on one or both spaces (manifolds) we can exploit this additional structure in order to show that the desired morphism must exist.

Rigidity theorem is about when a fairly weak equivalence between two manifolds (usually a homotopy equivalence) implies the existence of stronger equivalence homeomorphism, diffeomorphism or isometry.

Definition.

A closed topological manifold M is called topological rigid if any homotopy equivalence f : N → M with some manifold N as source and M as target is homotopic to a homeomorphism.

Examples

Example 1.
If closed 2-manifolds M and N are homotopically equivalent then they are homeomorphic. Moreover any homotopy equivalence of closed surfaces deforms to a homeomorphism.

Example 2.
If a closed manifold Mⁿ (n ≠ 3) is homotopy-equivalent to Sⁿ then Mⁿ is homeomorphic to Sⁿ.

Rigidity theorem in geometry

Definition.

A diffeomorphism of flat-Riemannian manifolds is said to be affine iff it carries geodesics to geodesic.

Theorem (Bieberbach)

If f : M → N is a homotopy equivalence between flat closed connected Riemannian manifolds then f is homotopic to an affine homeomorphism.

Mostow's rigidity theorem

Theorem: Let M and N be compact, locally symmetric Riemannian manifolds with everywhere non-positive curvature having no closed one or two dimensional geodesic subspace which are direct factor locally. If f : M → N is a homotopy equivalence then f is homotopic to an isometry.

Theorem (Mostow's theorem for hyperbolic n-manifolds, n ≥ 3): If M and N are complete hyperbolic n-manifolds, n ≥ 3 with finite volume and f : M → N is a homotopy equivalence then f is homotopic to an isometry.

These results are named after George Mostow.

Algebraic form

Let Γ and Δ be discrete subgroups of the isometry group of hyperbolic n-space H, where n ≥ 3, whose quotients H/Γ and H/Δ have finite volume. If Γ and Δ are isomorphic as discrete groups then they are conjugate.

Remarks

(1) In the 2-dimensional case any manifold of genus at least two has a hyperbolic structure. Mostow's rigidity theorem does not apply in this case. In fact, there are many hyperbolic structures on any such manifold; each such structure corresponds to a point in Teichmuller space.

(2) On the other hand if M and N are 2-manifolds of finite volume then it is easy to show that they are homeomorphic exactly when their fundamental groups are the same.

Application

The group of isometries of a finite-volume hyperbolic n-manifold M (for n ≥ 3) is finite and isomorphic to π₁(M).

References