Relative homology at a basepoint is reduced homology: 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

fully proved10Proved, and every prerequisite is fully proved too.

graph LR n0["Relative singular chains and homology"]:::fully_proved n1["Reduced singular homology"]:::fully_proved n2("The reduced long exact sequence of a nonempty pair"):::fully_proved n3("Homology of a point"):::fully_proved n4["Relative homology at a basepoint is reduced homology"]:::fully_proved n5("Good-pair relative homology is reduced quotient homology"):::fully_proved n6("The pointed relative-to-reduced comparison is natural"):::fully_proved n7("Reduced homology of a contractible space vanishes"):::fully_proved n8["The coproduct pair of a family of pointed spaces"]:::fully_proved n9("Reduced homology of a wedge is the direct sum of the summands"):::fully_proved n7 -.-> n9 n3 -.-> n4 n6 -.-> n9 n4 --> n8 n4 --> n9 n4 -.-> n7 n4 --> n5 n4 --> n6 n2 -.-> n4 n1 --> n7 n1 --> n4 n0 --> n4 click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-singular-homology" "Relative singular chains and homology — 1 item · 1 fully proved" click n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#reduced-singular-homology" "Reduced singular homology — 1 item · 1 fully proved" click n2 "../../../../../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 n3 "../../../../../roadmap/homology/simplicial-and-singular/index.html#point-homology" "Homology of a point — 1 item · 1 fully proved" class n4 focus click n4 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#pointed-relative-reduced-homology" "Relative homology at a basepoint is reduced homology — 1 item · 1 fully proved" click n5 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#good-pair-point-quotient-relative-homology" "Good-pair relative homology is reduced quotient homology — 1 item · 1 fully proved" click n6 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#pointed-relative-reduced-homology-naturality" "The pointed relative-to-reduced comparison is natural — 1 item · 1 fully proved" click n7 "../../../../../roadmap/homology/computations-and-applications/sphere-homology/index.html#contractible-space-reduced-homology" "Reduced homology of a contractible space vanishes — 1 item · 1 fully proved" click n8 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#sigma-pointed-pair" "The coproduct pair of a family of pointed spaces — 1 item · 1 fully proved" click n9 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#wedge-reduced-homology" "Reduced homology of a wedge is the direct sum of the summands — 1 item · 1 fully proved"