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Homotopic maps induce the same map on singular homology
Statement
If are homotopic continuous maps, then for every and every abelian group the induced homomorphisms on singular homology agree:
Facts & Assumptions
Given: A homotopy between continuous maps , an abelian group , and an integer .
The prism operator of a homotopy satisfies (The singular chain homotopy formula).
A family with is a chain homotopy (A chain homotopy).
Chain-homotopic chain maps induce the same map on homology (Chain-homotopic maps induce the same map on homology).
Proof
Extend the prism operator by on the zero group . By [L1], so the family satisfies the defining identity of [L2] for a chain homotopy from to . Thus the two induced singular chain maps are chain-homotopic.
Applying [L3] to the singular chain complex yields for every .
Depends on
Used by
- The Lefschetz number is a homotopy invariant Corollary
- A homology cobordism need not be an h-cobordism Counterexample
- The tautological line over RP∞ has no finite-rank complement Counterexample
- Algebraic Lefschetz number via rational homology traces Definition
- Lefschetz number of a finite CW self-map Definition
- Mobius band as an interval bundle with monodromy Example
- A closed three-manifold with H₁ = Z/2 and H₂ = 0 does not embed in S⁴ Lemma
- A collar constructs the relative orientation class and its boundary class Lemma
- A degree-zero sphere map extends over the ball without zeros Lemma
- Braid mapping classes lift equivariantly to the Burau cover Lemma
- Cap duality on a Euclidean coordinate ball Lemma
- Compatible orientation classes over compact subsets Lemma
- Coordinate-ball classes identify local homology stalks Lemma
- Rational cohomology scalar comparison on compact smooth manifolds Lemma
- The normal Thom class realizes the Poincare dual of a closed submanifold Lemma
- Degree is homotopy invariant and multiplicative under composition Proposition
- Degree-d sphere maps act by multiplication by d in any generalized theory Proposition
- Alexander duality for compact locally contractible subsets of a sphere Theorem
- Fully relative Poincaré–Lefschetz duality Theorem
- Homotopy equivalences induce isomorphisms on singular homology Theorem
- Integral homology comparison gives finite-range homotopy comparison for simply connected CW complexes Theorem
- Invariance of dimension for euclidean spaces Theorem
- Lefschetz-Hopf index formula Theorem
- Local homology detects manifold dimension, interior, and boundary Theorem
- Morse homology is naturally isomorphic to singular homology Theorem
- Poincaré–Lefschetz duality Theorem
- Singular homology satisfies homotopy exactness and excision Theorem
- The geometric intersection number is the Poincare-dual cup pairing Theorem
- Top homology of a connected manifold Theorem
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Allen Hatcher, Algebraic Topology (standard reference, not scraped)