Excision for singular homology

Depends on

• The cover-small inclusion is a chain homotopy equivalence
• Relative singular homology
• Functoriality of relative homology

Used by

• Good pairs and quotient reduced homology Corollary
• Homotopy spheres of dimension at least five bounding a contractible manifold are standard spheres Corollary
• Relative homology of a single handle pair Corollary
• Excision fails without closure inside interior Counterexample
• Fundamental class of a compact oriented manifold Definition
• Local degree at an isolated preimage Definition
• Orientation-preserving homeomorphisms and the geometric definition of quasiconformality Definition
• A collar constructs the relative orientation class and its boundary class Lemma
• Cap product and the Mayer–Vietoris duality ladder Lemma
• Collared gluing has relative excision and evaluation maps Lemma
• Compatible orientation classes over compact subsets Lemma
• Connected sum descends to oriented h-cobordism classes Lemma
• Connected sum preserves oriented homotopy spheres Lemma
• Coordinate-ball classes identify local homology stalks Lemma
• Local sphere orientations and finite puncture excision Lemma
• One handle changes relative homology in one degree only Lemma
• Orientation reversal is the connected-sum inverse Lemma
• Relative homology over one base cell is shifted fiber homology Lemma
• Smooth orientation sign is the local integral homology multiplier Lemma
• The inverse of a quasiconformal map is quasiconformal with the same dilatation Lemma
• The two-disk complement of a homotopy sphere is an h-cobordism Lemma
• Top de Rham integration equals fundamental-class evaluation Lemma
• Wall nonadditivity, RT cap references, and the Walker weight sign Lemma
• 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
• Cellular chains compute local homology Theorem
• Excision and Mayer–Vietoris with local coefficients Theorem
• Fully relative Poincaré–Lefschetz duality Theorem
• Local homology detects manifold dimension, interior, and boundary Theorem
• Morse homology is naturally isomorphic to singular homology Theorem
• Poincaré–Lefschetz duality Theorem
• Regular-value formula for degree Theorem
• Singular homology satisfies homotopy exactness and excision Theorem
• The geometric and analytic definitions of quasiconformality agree Theorem
• The symplectic period formula for integrals of wedge products Theorem
• Top homology of a connected manifold Theorem

Dependency tree · two levels

12 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, Theorem 2.20 (standard reference, not scraped)