Homotopy - Homotopy Invariance

Homotopy Invariance

Homotopy equivalence is important because in algebraic topology many concepts are homotopy invariant, that is, they respect the relation of homotopy equivalence. For example, if X and Y are homotopy equivalent spaces, then:

• If X is path-connected then so is Y.
• If X is simply connected then so is Y.
• The (singular) homology and cohomology groups of X and Y are isomorphic.
• If X and Y are path-connected, then the fundamental groups of X and Y are isomorphic, and so are the higher homotopy groups. (Without the path-connectedness assumption, one has π₁(X,x₀) isomorphic to π₁(Y,f(x₀)) where f : X → Y is a homotopy equivalence and x₀ ∈ X.)

An example of an algebraic invariant of topological spaces which is not homotopy-invariant is compactly supported homology (which is, roughly speaking, the homology of the compactification, and compactification is not homotopy-invariant).