The binary-cover excision chain map is a homotopy equivalence: local context
Project map · Chapter map · Full theorem DAG · Open textbook statement
This local map shows one dependency hop in either direction. The highlighted item is the current focus. fully proved7Proved, and every prerequisite is fully proved too.
graph LR
n0["Relative singular chains and homology"]:::fully_proved
n1["Small singular chains include by a chain-homotopy equivalence"]:::fully_proved
n2("Relative excision reduces to the union subcomplex"):::fully_proved
n3("The binary-cover excision chain map is a homotopy equivalence"):::fully_proved
n4["Binary-cover excision induces homology isomorphisms"]:::fully_proved
n5("The binary-cover Mayer–Vietoris sequence"):::fully_proved
n6["The mapping cone of a topological pair"]:::fully_proved
n0 --> n3
n0 --> n4
n3 --> n5
n3 --> n6
n3 --> n4
n2 -.-> n3
n1 -.-> n5
n1 -.-> n3
click n0 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-singular-homology" "Relative singular chains and homology — 1 item · 1 fully proved"
click n1 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#small-chain-inclusion-homotopy-equivalence" "Small singular chains include by a chain-homotopy equivalence — 1 item · 1 fully proved"
click n2 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#relative-chain-union-devissage" "Relative excision reduces to the union subcomplex — 1 item · 1 fully proved"
class n3 focus
click n3 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#binary-cover-excision-chain-equivalence" "The binary-cover excision chain map is a homotopy equivalence — 1 item · 1 fully proved"
click n4 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#binary-cover-excision-homology-isomorphism" "Binary-cover excision induces homology isomorphisms — 1 item · 1 fully proved"
click n5 "../../../../../../roadmap/homology/computations-and-applications/mayer-vietoris/index.html#binary-cover-mayer-vietoris-sequence" "The binary-cover Mayer–Vietoris sequence — 1 item · 1 fully proved"
click n6 "../../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#mapping-cone-model" "The mapping cone of a topological pair — 1 item · 1 fully proved"