The reduced long exact sequence of a nonempty 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. What the colours mean

fully proved9Proved, and every prerequisite is fully proved too.

graph LR n0("The long exact sequence of a pair"):::fully_proved n1["Reduced and ordinary homology agree in positive degrees"]:::fully_proved n2["Zeroth homology splits into reduced homology and the coefficient object"]:::fully_proved n3("The reduced long exact sequence of a nonempty pair"):::fully_proved n4["Relative homology at a basepoint is reduced homology"]:::fully_proved n5("The reduced pair long exact sequence is natural"):::fully_proved n6["A contractible subspace identifies reduced and relative homology"]:::fully_proved n7["Relative homology in a contractible ambient space"]:::fully_proved n8("The good-pair quotient exact sequence"):::fully_proved n0 --> n3 n2 --> n3 n3 --> n7 n3 -.-> n6 n3 --> n8 n3 -.-> n4 n3 --> n5 n1 --> n3 click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#pair-long-exact-sequence" "The long exact sequence of a pair — 1 item · 1 fully proved" click n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-positive-degree-comparison" "Reduced and ordinary homology agree in positive degrees — 1 item · 1 fully proved" click n2 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#pointed-zeroth-homology-splitting" "Zeroth homology splits into reduced homology and the coefficient object — 1 item · 1 fully proved" class n3 focus click n3 "../../../../../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 n4 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#pointed-relative-reduced-homology" "Relative homology at a basepoint is reduced homology — 1 item · 1 fully proved" click n5 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-pair-long-exact-sequence-naturality" "The reduced pair long exact sequence is natural — 1 item · 1 fully proved" click n6 "../../../../../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 n7 "../../../../../roadmap/homology/computations-and-applications/local-homology-and-dimension/index.html#contractible-ambient-relative-homology" "Relative homology in a contractible ambient space — 1 item · 1 fully proved" click n8 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#good-pair-point-quotient-exact-sequence" "The good-pair quotient exact sequence — 1 item · 1 fully proved"