Reduced homology of a wedge is the direct sum of the summands: 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 proved9Proved, and every prerequisite is fully proved too.

graph LR n0("The standard cover of a well-pointed wedge"):::fully_proved n1["Relative homology at a basepoint is reduced homology"]:::fully_proved n2("The pointed relative-to-reduced comparison is natural"):::fully_proved n3["The good-pair quotient comparison is natural"]:::fully_proved n4("Reduced homology of a contractible space vanishes"):::fully_proved n5["Relative homology commutes with exact coproducts"]:::fully_proved n6["The pointed wedge is the point quotient of the sigma pair"]:::fully_proved n7("A nonempty coproduct of pointed good pairs is good"):::fully_proved n8("Reduced homology of a wedge is the direct sum of the summands"):::fully_proved n0 -.-> n8 n6 --> n8 n5 --> n8 n7 --> n8 n4 -.-> n8 n3 -.-> n8 n2 -.-> n8 n2 --> n3 n1 --> n8 n1 -.-> n4 n1 --> n2 click n0 "../../../../../roadmap/fundamental-group/van-kampen/index.html#well-pointed-wedge-cover" "The standard cover of a well-pointed wedge — 1 item · 1 fully proved" click n1 "../../../../../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 n2 "../../../../../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 n3 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/good-pair-quotient/index.html#good-pair-quotient-comparison-naturality" "The good-pair quotient comparison is natural — 1 item · 1 fully proved" click n4 "../../../../../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 n5 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#relative-homology-coproduct" "Relative homology commutes with exact coproducts — 1 item · 1 fully proved" click n6 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#pointed-wedge-point-quotient" "The pointed wedge is the point quotient of the sigma pair — 1 item · 1 fully proved" click n7 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#sigma-good-pair" "A nonempty coproduct of pointed good pairs is good — 1 item · 1 fully proved" class n8 focus click n8 "../../../../../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"