The mapping-cone cover has the relative homology of the original 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 proved8Proved, and every prerequisite is fully proved too.
graph LR
n0["The base-side cone cover retracts onto the original space"]:::fully_proved
n1["The single-cone cover intersection has the homotopy type of its boundary"]:::fully_proved
n2["Relative singular chains and homology"]:::fully_proved
n3("Componentwise isomorphisms imply a relative isomorphism"):::fully_proved
n4["A homotopy equivalence induces homology isomorphisms"]:::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 -.-> n6
n1 -.-> n6
n6 --> n7
n5 --> n6
n5 --> n7
n4 -.-> n6
n3 -.-> n6
n2 --> n6
n2 --> n7
n2 --> n3
click n0 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#single-cone-base-retract" "The base-side cone cover retracts onto the original space — 1 item · 1 fully proved"
click n1 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#single-cone-intersection" "The single-cone cover intersection has the homotopy type of its boundary — 1 item · 1 fully proved"
click n2 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-singular-homology" "Relative singular chains and homology — 1 item · 1 fully proved"
click n3 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#relative-homology-map-isomorphism-criterion" "Componentwise isomorphisms imply a relative isomorphism — 1 item · 1 fully proved"
click n4 "../../../../../roadmap/homology/simplicial-and-singular/index.html#homotopy-equivalence-homology-iso" "A homotopy equivalence induces homology isomorphisms — 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"
class n6 focus
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"
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"