Relative chains of a triple form a short exact sequence: 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["Topological triples and their maps"]:::fully_proved
n1["Relative homology of a simplicial-set pair"]:::mathlib
n2["Relative singular chains and homology"]:::fully_proved
n3("Relative chains of a triple form a short exact sequence"):::fully_proved
n4("The long exact sequence of a triple"):::fully_proved
n5("The triple connecting map sends a relative cycle to its boundary"):::fully_proved
n2 --> n3
n1 -.-> n3
n0 --> n3
n3 --> n5
n3 --> n4
n4 -.-> n5
click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#topological-triple" "Topological triples and their maps — 1 item · 1 fully proved"
click n1 "../../../../../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 n2 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-singular-homology" "Relative singular chains and homology — 1 item · 1 fully proved"
class n3 focus
click n3 "../../../../../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"
click n4 "../../../../../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 n5 "../../../../../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"