A contractible subspace identifies reduced and relative homology: 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

fully proved9Proved, and every prerequisite is fully proved too.

graph LR n0["Relative singular chains and homology"]:::fully_proved n1["Reduced singular homology"]:::fully_proved n2("The reduced long exact sequence of a nonempty pair"):::fully_proved n3("The pointed relative-to-reduced comparison is natural"):::fully_proved n4("Reduced homology of a contractible space vanishes"):::fully_proved n5["A contractible subspace identifies reduced and relative homology"]:::fully_proved n6("The double-simplex difference generates reduced homology"):::fully_proved n7["The ordered sphere class orients local homology"]:::fully_proved n8["Relative homology is reduced homology of the mapping cone"]:::fully_proved n5 --> n7 n5 -.-> n6 n5 --> n8 n4 -.-> n5 n3 --> n5 n3 -.-> n6 n2 -.-> n5 n1 --> n5 n1 --> n8 n1 --> n4 n0 --> n5 n0 --> n8 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/index.html#reduced-singular-homology" "Reduced singular homology — 1 item · 1 fully proved" click n2 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-pair-long-exact-sequence" "The reduced long exact sequence of a nonempty pair — 1 item · 1 fully proved" click n3 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#pointed-relative-reduced-homology-naturality" "The pointed relative-to-reduced comparison is natural — 1 item · 1 fully proved" click n4 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#contractible-space-reduced-homology" "Reduced homology of a contractible space vanishes — 1 item · 1 fully proved" class n5 focus click n5 "../../../../../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 n6 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-fundamental-class" "The double-simplex difference generates reduced homology — 1 item · 1 fully proved" click n7 "../../../../../roadmap/homology/computations-and-applications/local-degree/index.html#sphere-local-orientation" "The ordered sphere class orients local homology — 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"