Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Naturality of the pair long exact sequence
Statement
A map of pairs induces a commuting morphism from the long exact sequence of to that of , including the connecting maps.
Facts & Assumptions
Given: A map of pairs .
Proof
The maps on , , and quotient chains give a commuting morphism of the short exact sequences used for the two pair sequences.
Naturality of the homological long exact sequence makes every resulting square commute. In particular, chainwise gives commutation at the connector.
Depends on
Used by
- Local degrees of a polynomial map on the riemann sphere Example
- A product collar deformation retracts onto its face Lemma
- Handle boundary coefficients are attaching-belt intersection numbers Lemma
- Local sphere orientations and finite puncture excision Lemma
- One handle changes relative homology in one degree only Lemma
- The cyclic cover retracts onto the lifted flower and has a deck-equivariant spine model Lemma
- The inverse of a quasiconformal map is quasiconformal with the same dilatation Lemma
- The relative handle chain complex computes H_*(W,M₀) and has the intersection matrix as its differential Proposition
- The unreduced module fits an exact sequence with the reduced module Proposition
- Singular homology satisfies homotopy exactness and excision Theorem
Dependency tree · two levels
9 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)