The long exact sequence of a triple: 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 proved5Proved, and every prerequisite is fully proved too.
graph LR
n0("A short exact sequence gives an exact homology sequence"):::mathlib
n1("Relative chains of a triple form a short exact sequence"):::fully_proved
n2("The long exact sequence of a triple"):::fully_proved
n3("The triple connecting map sends a relative cycle to its boundary"):::fully_proved
n4("The identity simplex generates relative simplex homology"):::fully_proved
n5("The triple long exact sequence is natural"):::fully_proved
n3 -.-> n4
n0 -.-> n2
n0 -.-> n5
n1 --> n3
n1 --> n2
n2 -.-> n4
n2 -.-> n3
n2 --> n5
click n0 "../../../../../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 n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#triple-chain-short-exact-sequence" "Relative chains of a triple form a short exact sequence — 1 item · 1 fully proved"
class n2 focus
click n2 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#triple-long-exact-sequence" "The long exact sequence of a triple — 1 item · 1 fully proved"
click n3 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#triple-connecting-cycle-formula" "The triple connecting map sends a relative cycle to its boundary — 1 item · 1 fully proved"
click n4 "../../../../../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 n5 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#triple-long-exact-sequence-naturality" "The triple long exact sequence is natural — 1 item · 1 fully proved"