Relative singular homology

Depends on

• Relative singular chain complex
• Boundary on relative chains

Used by

• A homology cobordism need not be an h-cobordism Counterexample
• Based cellular chains of a universal cover as finite free right group-ring modules Definition
• Poincare polynomial of a space and of a pair over a field Definition
• Relative cap products with quotient domains displayed Definition
• The unreduced Burau relative homology module Definition
• Relative Morse homology of a single-handle cobordism Example
• A contractible relative group-ring complex with a pi-one isomorphism detects a homotopy equivalence Lemma
• A multiple of a fork surface has a closed compact replacement Lemma
• A product collar deformation retracts onto its face Lemma
• Attaching handles of index at least q preserves homology below q-1 Lemma
• Braid mapping classes lift equivariantly to the Burau cover Lemma
• Cellular reduction for a highly connected pair Lemma
• Collared gluing has relative excision and evaluation maps Lemma
• Compatible orientation classes over compact subsets Lemma
• Cubical pinch is additive on relative homology Lemma
• Long exact sequence of a triple in singular homology Lemma
• One handle changes relative homology in one degree only Lemma
• Properties of bounded-domain degree Lemma
• Relative homology Mayer–Vietoris for closed supports Lemma
• Relative homology of an h-cobordism vanishes at both ends Lemma
• Relative Kronecker evaluation is well defined, biadditive and natural Lemma
• Serre-fibration replacement preserves fiber homology transport Lemma
• The fork-noodle pairing is well defined and equivariant Lemma
• The geometric half twist acts on the lifted-edge basis by the Burau block Lemma
• Universal-cover boundaries, maps and homotopies respect the right group-ring action Lemma
• Weak homotopy equivalences induce integral homology isomorphisms without choice Lemma
• Well-definedness of the relative connector Lemma
• Functoriality of relative homology Proposition
• Relative homology for the empty and total subspace Proposition
• Relative Morse inequalities for a cobordism Proposition
• The handle chain complex computes singular homology Proposition
• The homology effect of surgery away from the middle dimensions Proposition
• The relative handle chain complex computes H_*(W,M₀) and has the intersection matrix as its differential Proposition
• The relative Morse complex of an adapted cobordism Proposition
• The unreduced module fits an exact sequence with the reduced module Proposition
• Composition and based-pair sum formulas for Whitehead torsion Theorem
• Excision for singular homology Theorem
• Homological Serre spectral sequence Theorem
• Long exact sequence of a pair Theorem
• Morse polynomial identity Theorem

…and 1 more result.

Dependency tree · two levels

4 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, §2.1 (standard reference, not scraped)