Homotopy Invariance of Homology

Algebraic Topology · Axiom Academy

LESSON Homotopy Invariance of Homology Homology depends only on homotopy type, not continuous structure Two continuous maps f, g: X → Y are homotopic if there exists a continuous family of maps connecting them. Intuitively, we can continuously deform f into g through intermediate maps H(-,t). This is an equivalence relation on the set of continuous maps from X to Y. 2. Homotopic Maps Induce Equal Homomorphisms The fundamental theorem: if f and g are homotopic, then they induce the same homomorphism on all homology groups. This is proven by constructing a chain homotopy: an algebraic analogue of topological homotopy. The key is showing that the difference f_# - g_# equals ∂P + P∂ for some operator P, which vanishes on homology. 3. Homotopy Equivalence and Homology Homotopy equivalent spaces have isomorphic homology groups. This makes homology a homotopy invariant. Example: Any contractible space (homotopy equivalent to a point) has trivial homology except in dimension 0: H_0(X) = ℤ and H_n(X) = 0 for n > 0. This includes disks, solid balls, and all of Euclidean space.

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