Reduced and ordinary homology agree in positive degrees: 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["Singular homology"]:::mathlib n1["Reduced singular homology"]:::fully_proved n2["Reduced and ordinary homology agree in positive degrees"]:::fully_proved n3("The reduced long exact sequence of a nonempty pair"):::fully_proved n4["The ordered class orients ordinary sphere homology"]:::fully_proved n5("Reduced homology of the zero-sphere"):::fully_proved n2 --> n4 n2 -.-> n5 n2 --> n3 n1 --> n5 n1 --> n2 n0 --> n2 click n0 "../../../../../roadmap/homology/simplicial-and-singular/index.html#singular-homology" "Singular homology — 1 item · 1 in mathlib" click n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-singular-homology" "Reduced singular homology — 1 item · 1 fully proved" class n2 focus click n2 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-positive-degree-comparison" "Reduced and ordinary homology agree in positive degrees — 1 item · 1 fully proved" click n3 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-pair-long-exact-sequence" "The reduced long exact sequence of a nonempty pair — 1 item · 1 fully proved" click n4 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#integral-sphere-orientation" "The ordered class orients ordinary sphere homology — 1 item · 1 fully proved" click n5 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#zero-sphere-reduced-homology" "Reduced homology of the zero-sphere — 1 item · 1 fully proved"