The augmented singular chain complex: 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. What the colours mean

in mathlib1Upstreamed into Mathlib.fully proved5Proved, and every prerequisite is fully proved too.

graph LR n0["The singular chain complex"]:::mathlib n1["The augmented singular chain complex"]:::fully_proved n2["Reduced singular homology"]:::fully_proved n3("The reduced pair long exact sequence is natural"):::fully_proved n4["Small augmented chains include by a chain-homotopy equivalence"]:::fully_proved n5("Augmented binary-cover chains form a short exact sequence"):::fully_proved n4 --> n5 n1 --> n5 n1 --> n4 n1 -.-> n3 n1 --> n2 n0 --> n1 click n0 "../../../../../roadmap/homology/simplicial-and-singular/index.html#singular-chain-complex" "The singular chain complex — 1 item · 1 in mathlib" class n1 focus click n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#augmented-singular-chain-complex" "The augmented singular chain complex — 1 item · 1 fully proved" click n2 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-singular-homology" "Reduced singular homology — 1 item · 1 fully proved" click n3 "../../../../../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 n4 "../../../../../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" click n5 "../../../../../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"