Relative homology is reduced homology of the mapping cone: 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 proved8Proved, and every prerequisite is fully proved too.
graph LR
n0("The cone-side cover member is contractible"):::fully_proved
n1["Relative singular chains and homology"]:::fully_proved
n2["Reduced singular homology"]:::fully_proved
n3["Binary-cover excision induces homology isomorphisms"]:::fully_proved
n4["A contractible subspace identifies reduced and relative homology"]:::fully_proved
n5["The mapping cone of a topological pair"]:::fully_proved
n6["The mapping-cone cover has the relative homology of the original pair"]:::fully_proved
n7["Relative homology is reduced homology of the mapping cone"]:::fully_proved
n0 -.-> n7
n4 --> n7
n6 --> n7
n5 --> n6
n5 --> n7
n2 --> n4
n2 --> n7
n1 --> n4
n1 --> n6
n1 --> n7
n1 --> n3
n3 -.-> n7
click n0 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#single-cone-upper-contractible" "The cone-side cover member is contractible — 1 item · 1 fully proved"
click n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-singular-homology" "Relative singular chains and homology — 1 item · 1 fully proved"
click n2 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-singular-homology" "Reduced singular homology — 1 item · 1 fully proved"
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/computations-and-applications/post-excision-applications/index.html#contractible-subspace-relative-reduced-homology" "A contractible subspace identifies reduced and relative homology — 1 item · 1 fully proved"
click n5 "../../../../../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"
click n6 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#mapping-cone-cover-relative-homology" "The mapping-cone cover has the relative homology of the original pair — 1 item · 1 fully proved"
class n7 focus
click n7 "../../../../../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"