Augmented binary-cover chains 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. fully proved6Proved, and every prerequisite is fully proved too.
graph LR
n0["The augmented singular chain complex"]:::fully_proved
n1["The binary-cover Mayer–Vietoris chain complex"]:::fully_proved
n2("Binary-cover chains form a short exact sequence"):::fully_proved
n3["Small augmented chains include by a chain-homotopy equivalence"]:::fully_proved
n4("Augmented binary-cover chains form a short exact sequence"):::fully_proved
n5("The reduced binary-cover Mayer–Vietoris sequence"):::fully_proved
n4 --> n5
n1 --> n4
n1 --> n2
n2 -.-> n4
n3 --> n4
n3 -.-> n5
n0 --> n4
n0 --> n3
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/computations-and-applications/mayer-vietoris/index.html#binary-cover-chain-complex" "The binary-cover Mayer–Vietoris chain complex — 1 item · 1 fully proved"
click n2 "../../../../../roadmap/homology/computations-and-applications/mayer-vietoris/index.html#binary-cover-chain-short-exact-sequence" "Binary-cover chains form a short exact sequence — 1 item · 1 fully proved"
click n3 "../../../../../roadmap/homology/computations-and-applications/mayer-vietoris/index.html#small-augmented-chain-inclusion-homotopy-equivalence" "Small augmented chains include by a chain-homotopy equivalence — 1 item · 1 fully proved"
class n4 focus
click n4 "../../../../../roadmap/homology/computations-and-applications/mayer-vietoris/index.html#augmented-binary-cover-chain-short-exact-sequence" "Augmented binary-cover chains form a short exact sequence — 1 item · 1 fully proved"
click n5 "../../../../../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"