The fundamental group of the circle: 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 proved14Proved, and every prerequisite is fully proved too.

graph LR n0("The real line covers the circle"):::fully_proved n1["Winding number of a loop in the circle"]:::fully_proved n2("The fundamental group of the circle"):::fully_proved n3("Borsuk–Ulam for the two-sphere"):::fully_proved n4("The circle is not a retract of the disc"):::fully_proved n5("Euclidean two-space is distinguished by dimension"):::fully_proved n6("The fundamental theorem of algebra"):::fully_proved n7["The fundamental group of the torus"]:::fully_proved n8["The boundary of the two-disk is the circle"]:::fully_proved n9("Overlap meridians recover the attaching loops"):::fully_proved n10["Attaching 2-cells adds the attaching relations"]:::fully_proved n11("The degree-n circle map sends the generator to the degree-n loop"):::fully_proved n12["The degree-n circle relation gives the cyclic group"]:::fully_proved n13["The fundamental group of a wedge of circles"]:::fully_proved n0 -.-> n2 n0 --> n1 n2 --> n3 n2 --> n5 n2 --> n6 n2 --> n4 n2 --> n7 n2 --> n10 n2 -.-> n9 n2 --> n11 n2 --> n12 n2 --> n8 n2 --> n13 n1 --> n2 n9 --> n10 n8 --> n10 n8 --> n9 click n0 "../../../../roadmap/fundamental-group/basic-constructions/index.html#circle-covering" "The real line covers the circle — 1 item · 1 fully proved" click n1 "../../../../roadmap/fundamental-group/basic-constructions/index.html#winding-number" "Winding number of a loop in the circle — 1 item · 1 fully proved" class n2 focus click n2 "../../../../roadmap/fundamental-group/basic-constructions/index.html#fundamental-group-circle" "The fundamental group of the circle — 1 item · 1 fully proved" click n3 "../../../../roadmap/fundamental-group/basic-constructions/index.html#borsuk-ulam-sphere" "Borsuk–Ulam for the two-sphere — 1 item · 1 fully proved" click n4 "../../../../roadmap/fundamental-group/basic-constructions/index.html#no-retraction-disc" "The circle is not a retract of the disc — 1 item · 1 fully proved" click n5 "../../../../roadmap/fundamental-group/basic-constructions/index.html#euclidean-two-not-homeomorphic" "Euclidean two-space is distinguished by dimension — 1 item · 1 fully proved" click n6 "../../../../roadmap/fundamental-group/basic-constructions/index.html#fundamental-theorem-algebra" "The fundamental theorem of algebra — 1 item · 1 fully proved" click n7 "../../../../roadmap/fundamental-group/basic-constructions/index.html#torus-fundamental-group" "The fundamental group of the torus — 1 item · 1 fully proved" click n8 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#disk-boundary-two-circle" "The boundary of the two-disk is the circle — 1 item · 1 fully proved" click n9 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/auxiliary/index.html#auxiliary-overlap-attaching-map" "Overlap meridians recover the attaching loops — 1 item · 1 fully proved" click n10 "../../../../roadmap/fundamental-group/van-kampen/index.html#attach-two-cells-fundamental-group" "Attaching 2-cells adds the attaching relations — 1 item · 1 fully proved" click n11 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#circle-degree-map" "The degree-n circle map sends the generator to the degree-n loop — 1 item · 1 fully proved" click n12 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#cyclic-relation-quotient" "The degree-n circle relation gives the cyclic group — 1 item · 1 fully proved" click n13 "../../../../roadmap/fundamental-group/van-kampen/index.html#wedge-circles-free-group" "The fundamental group of a wedge of circles — 1 item · 1 fully proved"