Binary-cover excision induces homology isomorphisms: 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. What the colours mean

in mathlib1Upstreamed into Mathlib.fully proved8Proved, and every prerequisite is fully proved too.

graph LR n0["Relative singular chains and homology"]:::fully_proved n1("Chain-homotopic maps induce the same homology map"):::mathlib n2("The binary-cover excision chain map is a homotopy equivalence"):::fully_proved n3["Binary-cover excision induces homology isomorphisms"]:::fully_proved n4["Deleted-subset excision induces homology isomorphisms"]:::fully_proved n5("The zero face induces an isomorphism in relative homology"):::fully_proved n6("Deleted-pair homology is local on an open neighborhood"):::fully_proved n7("Excision holds for a union of CW subcomplexes"):::fully_proved n8["Relative homology is reduced homology of the mapping cone"]:::fully_proved n1 -.-> n3 n0 --> n6 n0 --> n7 n0 --> n8 n0 --> n2 n0 --> n3 n0 --> n4 n2 --> n3 n3 -.-> n6 n3 -.-> n7 n3 -.-> n8 n3 -.-> n5 n3 -.-> n4 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/index.html#chain-homotopy-invariance" "Chain-homotopic maps induce the same homology map — 1 item · 1 in mathlib" click n2 "../../../../../../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" class n3 focus click n3 "../../../../../../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 n4 "../../../../../../roadmap/homology/simplicial-and-singular/relative-homology/small-chains-and-excision/index.html#deleted-subset-excision-homology-isomorphism" "Deleted-subset excision induces homology isomorphisms — 1 item · 1 fully proved" click n5 "../../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-zero-face-relative-homology" "The zero face induces an isomorphism in relative homology — 1 item · 1 fully proved" click n6 "../../../../../../roadmap/homology/computations-and-applications/local-degree/index.html#deleted-pair-open-neighborhood" "Deleted-pair homology is local on an open neighborhood — 1 item · 1 fully proved" click n7 "../../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#cw-subcomplex-excision" "Excision holds for a union of CW subcomplexes — 1 item · 1 fully proved" click n8 "../../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#mapping-cone-relative-reduced-homology" "Relative homology is reduced homology of the mapping cone — 1 item · 1 fully proved"