The reduced pair long exact sequence is natural: 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. in mathlib1Upstreamed into Mathlib.fully proved6Proved, and every prerequisite is fully proved too.
graph LR
n0["The augmented singular chain complex"]:::fully_proved
n1("A short exact sequence gives an exact homology sequence"):::mathlib
n2("The reduced long exact sequence of a nonempty pair"):::fully_proved
n3("The pair long exact sequence is natural"):::fully_proved
n4("The reduced pair long exact sequence is natural"):::fully_proved
n5("The pointed relative-to-reduced comparison is natural"):::fully_proved
n6("Naturality of the good-pair quotient sequence"):::fully_proved
n0 -.-> n4
n5 -.-> n6
n3 -.-> n4
n2 --> n4
n4 -.-> n6
n4 -.-> n5
n1 -.-> n3
n1 -.-> n4
click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#augmented-singular-chain-complex" "The augmented singular chain complex — 1 item · 1 fully proved"
click n1 "../../../../../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"
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/index.html#pair-long-exact-sequence-naturality" "The pair long exact sequence is natural — 1 item · 1 fully proved"
class n4 focus
click n4 "../../../../../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 n5 "../../../../../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 n6 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#good-pair-point-quotient-sequence-naturality" "Naturality of the good-pair quotient sequence — 1 item · 1 fully proved"