In the mathematical field of

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

''M'' is called topologically rigid if every manifold homotopically equivalent to ''M'' is also

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

to ''M''.

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 In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...

) implies the existence of stronger equivalence homeomorphism,

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

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 ''M''^''n'' (''n'' ≠ 3) is homotopy-equivalent to ''S''^''n'' then Mⁿ is homeomorphic to ''S''^''n''.

Rigidity theorems in geometry

Definition.

A diffeomorphism of flat-Riemannian manifolds is said to be affine

iff In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both ...

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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

, locally symmetric

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

s 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 George Daniel Mostow (July 4, 1923 – April 4, 2017) was an American mathematician, renowned for his contributions to Lie theory. He was the Henry Ford II (emeritus) Professor of Mathematics at Yale University, a member of the National Academy of ...

Algebraic form

Let Γ and Δ be discrete subgroups of the

isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element ...

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 finitely generated and isomorphic to π₁(''M'').

References

{{reflist Topology Maps of manifolds Homotopy theory