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Cellular maps induce cellular chain maps
Statement
For every abelian group , a cellular map , meaning , induces a chain map . Assuming the Axiom of Choice, its map on cellular homology corresponds to on singular homology under the natural cellular comparison for arbitrary CW complexes. If and are finite CW complexes, this compatibility holds without the Axiom of Choice.
Facts & Assumptions
Given: An abelian group and a cellular map . Assume the Axiom of Choice only for the arbitrary-CW singular-homology comparison.
Proof
The restrictions induce maps of relative groups, hence maps on cellular chains.
Naturality of pair connecting maps makes these maps commute with the differentials, without a choice assumption. Under the Axiom of Choice, Cellular homology computes singular homology compares cellular and singular homology naturally for arbitrary CW complexes, so the induced maps agree with . For finite and , use the finite skeletal exact-sequence calculations in that theorem, followed by finitely many skeletal stabilization isomorphisms: above the last cell dimension each space equals its final skeleton, so no infinite colimit is needed. Every pair and inclusion map in these finite calculations commutes with , giving the same compatibility without choice.
Depends on
Used by
- Algebraic Lefschetz number via rational homology traces Definition
- Cohomology of lens spaces from UCT Example
- Equal additive cohomology but different rings Example
- Integral cohomology ring of complex projective space Example
- Mod-two cohomology ring of real projective space Example
- The Lefschetz number of the identity is Euler characteristic Example
- Finite-cellular cyclic squares, Cartan formula, and cyclic-basis action Lemma
- Mod-two cohomology ring of infinite real projective space Lemma
- Mod-two cohomology rings of complex projective spaces Lemma
- Rational Hurewicz for arbitrary wedges of high-dimensional spheres Lemma
- Real projective space cellular homology and the pinch map Lemma
- Torsion Eilenberg–Mac Lane spaces are rationally acyclic Lemma
- Lefschetz fixed-point theorem for finite complexes Theorem
Dependency tree · two levels
13 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, Section 2.2 (standard reference, not scraped)