The antipodal map has degree determined by dimension: 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("Coordinate sign changes on the sphere"):::fully_proved n1("The formal properties of degree"):::fully_proved n2("A sphere reflection has degree minus one"):::fully_proved n3("The antipodal map has degree determined by dimension"):::fully_proved n4("A fixed-point-free sphere map has antipodal degree"):::fully_proved n5("A nonvanishing tangent field forces odd dimension"):::fully_proved n3 -.-> n4 n3 -.-> n5 n1 -.-> n3 n1 -.-> n2 n1 -.-> n4 n1 -.-> n5 n2 -.-> n3 n0 --> n3 n0 --> n2 click n0 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#sphere-coordinate-sign-changes" "Coordinate sign changes on the sphere — 1 item · 1 fully proved" click n1 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#basic-degree-properties" "The formal properties of degree — 1 item · 1 fully proved" click n2 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#coordinate-reflection-degree" "A sphere reflection has degree minus one — 1 item · 1 fully proved" class n3 focus click n3 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#antipodal-degree" "The antipodal map has degree determined by dimension — 1 item · 1 fully proved" click n4 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#fixed-point-free-degree" "A fixed-point-free sphere map has antipodal degree — 1 item · 1 fully proved" click n5 "../../../../../roadmap/homology/computations-and-applications/degree/index.html#tangent-field-parity-obstruction" "A nonvanishing tangent field forces odd dimension — 1 item · 1 fully proved"