The double-simplex difference generates reduced 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 proved8Proved, and every prerequisite is fully proved too.

graph LR n0["The ordered difference of the two simplices is a cycle"]:::fully_proved n1("The pointed relative-to-reduced comparison is natural"):::fully_proved n2("The identity simplex generates relative simplex homology"):::fully_proved n3("The first simplex computes homology relative to the second"):::fully_proved n4["A contractible subspace identifies reduced and relative homology"]:::fully_proved n5("The double-simplex difference generates reduced homology"):::fully_proved n6("Swapping the double simplex negates its fundamental class"):::fully_proved n7["The double simplex is a sphere"]:::fully_proved n4 -.-> n5 n5 --> n6 n5 --> n7 n0 --> n6 n0 --> n5 n0 --> n3 n0 --> n7 n3 -.-> n5 n2 -.-> n5 n1 --> n4 n1 -.-> n5 click n0 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-fundamental-cycle" "The ordered difference of the two simplices is a cycle — 1 item · 1 fully proved" click n1 "../../../../../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 n2 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#relative-simplex-fundamental-class" "The identity simplex generates relative simplex homology — 1 item · 1 fully proved" click n3 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-relative-comparison" "The first simplex computes homology relative to the second — 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" class n5 focus click n5 "../../../../../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 n6 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#double-simplex-swap-class" "Swapping the double simplex negates its fundamental class — 1 item · 1 fully proved" click n7 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-sphere-homeomorphism" "The double simplex is a sphere — 1 item · 1 fully proved"