Coordinate sign changes on the sphere: 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 proved6Proved, and every prerequisite is fully proved too.

graph LR n0("Swapping the double simplex negates its fundamental class"):::fully_proved n1["The double simplex is a sphere"]:::fully_proved n2("Coordinate sign changes on the sphere"):::fully_proved n3("A sphere reflection has degree minus one"):::fully_proved n4("The antipodal map has degree determined by dimension"):::fully_proved n5("Only the antipodal group acts freely on an even sphere"):::fully_proved n3 -.-> n4 n0 --> n2 n2 --> n4 n2 --> n3 n2 --> n5 n1 --> n2 click n0 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#double-simplex-swap-class" "Swapping the double simplex negates its fundamental class — 1 item · 1 fully proved" click n1 "../../../../../roadmap/homology/computations-and-applications/post-excision-applications/index.html#double-simplex-sphere-homeomorphism" "The double simplex is a sphere — 1 item · 1 fully proved" class n2 focus click n2 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#sphere-coordinate-sign-changes" "Coordinate sign changes on the sphere — 1 item · 1 fully proved" click n3 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#coordinate-reflection-degree" "A sphere reflection has degree minus one — 1 item · 1 fully proved" click n4 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#antipodal-degree" "The antipodal map has degree determined by dimension — 1 item · 1 fully proved" click n5 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#free-even-sphere-action" "Only the antipodal group acts freely on an even sphere — 1 item · 1 fully proved"