The long exact sequence of a 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

in mathlib3Upstreamed into Mathlib.fully proved5Proved, and every prerequisite is fully proved too.

graph LR n0["Relative homology of a simplicial-set pair"]:::mathlib n1["Singular homology"]:::mathlib n2["Relative singular chains and homology"]:::fully_proved n3("A short exact sequence gives an exact homology sequence"):::mathlib n4("The long exact sequence of a pair"):::fully_proved n5("The reduced long exact sequence of a nonempty pair"):::fully_proved n6("Componentwise isomorphisms imply a relative isomorphism"):::fully_proved n7("The pair long exact sequence is natural"):::fully_proved n4 -.-> n6 n4 --> n7 n4 --> n5 n2 --> n6 n2 --> n4 n2 --> n7 n3 -.-> n4 n3 -.-> n7 n0 -.-> n4 n1 --> n4 n1 --> n2 click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#simplicial-pair-relative-homology" "Relative homology of a simplicial-set pair — 1 item · 1 in mathlib" click n1 "../../../../../roadmap/homology/simplicial-and-singular/index.html#singular-homology" "Singular homology — 1 item · 1 in mathlib" 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/index.html#short-exact-homology-sequence" "A short exact sequence gives an exact homology sequence — 1 item · 1 in mathlib" class n4 focus click n4 "../../../../../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 n5 "../../../../../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 n6 "../../../../../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 n7 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#pair-long-exact-sequence-naturality" "The pair long exact sequence is natural — 1 item · 1 fully proved"