The boundary and zero horn of a standard topological simplex: 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["Topological triples and their maps"]:::fully_proved n1["Relative singular chains and homology"]:::fully_proved n2["The boundary and zero horn of a standard topological simplex"]:::fully_proved n3["The ordered difference of the two simplices is a cycle"]:::fully_proved n4["A standard simplex strongly deformation retracts onto its zero horn"]:::fully_proved n5["Standard-simplex boundary and horn pairs are good"]:::fully_proved n6["The zero face identifies the two simplex point quotients"]:::fully_proved n7("The zero face induces an isomorphism in relative homology"):::fully_proved n8("The identity simplex generates relative simplex homology"):::fully_proved n9["The standard simplex pair is homeomorphic to the disk pair"]:::fully_proved n5 -.-> n7 n2 --> n3 n2 --> n8 n2 --> n5 n2 --> n9 n2 --> n6 n2 --> n7 n2 --> n4 n6 -.-> n7 n7 -.-> n8 n4 -.-> n8 n1 --> n2 n0 --> n2 click n0 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#topological-triple" "Topological triples and their maps — 1 item · 1 fully proved" click n1 "../../../../../roadmap/homology/simplicial-and-singular/relative-homology/index.html#relative-singular-homology" "Relative singular chains and homology — 1 item · 1 fully proved" class n2 focus click n2 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-boundary-horn" "The boundary and zero horn of a standard topological simplex — 1 item · 1 fully proved" click n3 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-fundamental-cycle" "The ordered difference of the two simplices is a cycle — 1 item · 1 fully proved" click n4 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-zero-horn-retraction" "A standard simplex strongly deformation retracts onto its zero horn — 1 item · 1 fully proved" click n5 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-boundary-good-pairs" "Standard-simplex boundary and horn pairs are good — 1 item · 1 fully proved" click n6 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-zero-face-point-quotient" "The zero face identifies the two simplex point quotients — 1 item · 1 fully proved" click n7 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-zero-face-relative-homology" "The zero face induces an isomorphism in relative homology — 1 item · 1 fully proved" click n8 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#relative-simplex-fundamental-class" "The identity simplex generates relative simplex homology — 1 item · 1 fully proved" click n9 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#standard-simplex-disk-pair-homeomorphism" "The standard simplex pair is homeomorphic to the disk pair — 1 item · 1 fully proved"