The mapping cone of a topological pair: 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 proved6Proved, and every prerequisite is fully proved too.
graph LR
n0("A single cone attachment has a two-set open cover"):::fully_proved
n1("A single cone attachment is a topological pushout"):::fully_proved
n2("The binary-cover excision chain map is a homotopy equivalence"):::fully_proved
n3["The mapping cone of a topological pair"]:::fully_proved
n4["The mapping-cone cover has the relative homology of the original pair"]:::fully_proved
n5["Relative homology is reduced homology of the mapping cone"]:::fully_proved
n0 --> n1
n0 --> n3
n1 -.-> n3
n4 --> n5
n3 --> n4
n3 --> n5
n2 --> n3
click n0 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#single-cone-open-cover" "A single cone attachment has a two-set open cover — 1 item · 1 fully proved"
click n1 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#single-cone-pushout" "A single cone attachment is a topological pushout — 1 item · 1 fully proved"
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/computations-and-applications/post-excision-applications/index.html#mapping-cone-model" "The mapping cone of a topological pair — 1 item · 1 fully proved"
click n4 "../../../../../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"
click n5 "../../../../../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"