A short exact sequence gives an exact homology 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 mathlib2Upstreamed into Mathlib.fully proved7Proved, and every prerequisite is fully proved too.
graph LR
n0["A chain map induces a map on homology"]:::mathlib
n1("A short exact sequence gives an exact homology sequence"):::mathlib
n2("The long exact sequence of a 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 long exact sequence of a triple"):::fully_proved
n6("The binary-cover Mayer–Vietoris sequence"):::fully_proved
n7("The reduced binary-cover Mayer–Vietoris sequence"):::fully_proved
n8("The triple long exact sequence is natural"):::fully_proved
n0 --> n1
n2 --> n3
n3 -.-> n4
n1 -.-> n6
n1 -.-> n7
n1 -.-> n2
n1 -.-> n3
n1 -.-> n4
n1 -.-> n5
n1 -.-> n8
n5 --> n8
click n0 "../../../../../roadmap/homology/simplicial-and-singular/index.html#chain-map-homology" "A chain map induces a map on homology — 1 item · 1 in mathlib"
class n1 focus
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#pair-long-exact-sequence" "The long exact sequence of a 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"
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/index.html#triple-long-exact-sequence" "The long exact sequence of a triple — 1 item · 1 fully proved"
click n6 "../../../../../roadmap/homology/computations-and-applications/mayer-vietoris/index.html#binary-cover-mayer-vietoris-sequence" "The binary-cover Mayer–Vietoris sequence — 1 item · 1 fully proved"
click n7 "../../../../../roadmap/homology/computations-and-applications/mayer-vietoris/index.html#reduced-binary-cover-mayer-vietoris-sequence" "The reduced binary-cover Mayer–Vietoris sequence — 1 item · 1 fully proved"
click n8 "../../../../../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"