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Mayer–Vietoris sequence in singular homology
Statement
For an open cover , the sequence is exact, with middle map .
Facts & Assumptions
Given: An open cover .
Proof
Apply the homological long exact sequence to the short exact sequence of the preceding theorem.
The cover-small complex has the same homology as , so replacing its homology term identifies that long exact sequence with the displayed one.
Depends on
Used by
- Suspension isomorphism in reduced singular homology Corollary
- A homology cobordism need not be an h-cobordism Counterexample
- Mayer–Vietoris connecting class Definition
- Weighted Lagrangian ribbon bordisms Definition
- Mayer–Vietoris computation of first homology of the torus Example
- Mayer–Vietoris computation of sphere homology Example
- A closed three-manifold with H₁ = Z/2 and H₂ = 0 does not embed in S⁴ Lemma
- Cap product and the Mayer–Vietoris duality ladder Lemma
- Connected sum descends to oriented h-cobordism classes Lemma
- Connected sum preserves oriented homotopy spheres Lemma
- Projective-plane stabilization gives a five-bordism with the prescribed three-boundary Lemma
- Spherical leaf stability on a closed manifold needs only countable choice Lemma
- The double cover branched over a slice disk is a rational homology ball Lemma
- The two-disk complement of a homotopy sphere is an h-cobordism Lemma
- Wall nonadditivity, RT cap references, and the Walker weight sign Lemma
- Finiteness and additivity of the Euler characteristic Proposition
- Excision and Mayer–Vietoris with local coefficients Theorem
- Naturality of singular Mayer–Vietoris 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, §2.2 (standard reference, not scraped)