Attaching 2-cells adds the attaching relations: 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 proved13Proved, and every prerequisite is fully proved too.

graph LR n0("The fundamental group of the circle"):::fully_proved n1("Retractions and deformation retracts on the fundamental group"):::fully_proved n2["The auxiliary cover basepoint transports to the original basepoint"]:::fully_proved n3("An open-cover model for attached cells"):::fully_proved n4["An attachment with no cells is an isomorphism"]:::fully_proved n5("The attaching-sphere pieces generate the intersection group"):::fully_proved n6["The boundary of the two-disk is the circle"]:::fully_proved n7("Overlap meridians recover the attaching loops"):::fully_proved n8["A binary cover with a trivial second fundamental group"]:::fully_proved n9["Attaching 2-cells adds the attaching relations"]:::fully_proved n10["Free-group relators form a standard two-cell attachment"]:::fully_proved n11["The presentation complex has the presented fundamental group"]:::fully_proved n12["The cyclic presentation complex"]:::fully_proved n0 --> n9 n0 -.-> n7 n0 --> n6 n1 --> n9 n9 --> n12 n9 --> n11 n9 --> n10 n5 --> n9 n3 --> n9 n3 --> n5 n2 --> n9 n2 -.-> n3 n2 --> n7 n7 --> n9 n8 --> n9 n6 --> n9 n6 --> n7 n6 --> n12 n4 --> n9 n10 --> n11 click n0 "../../../../roadmap/fundamental-group/basic-constructions/index.html#fundamental-group-circle" "The fundamental group of the circle — 1 item · 1 fully proved" click n1 "../../../../roadmap/fundamental-group/basic-constructions/index.html#retractions-fundamental-group" "Retractions and deformation retracts on the fundamental group — 1 item · 1 fully proved" click n2 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/auxiliary/index.html#auxiliary-cover-basepoint-transport" "The auxiliary cover basepoint transports to the original basepoint — 1 item · 1 fully proved" click n3 "../../../../roadmap/fundamental-group/van-kampen/index.html#cell-attachment-cover-model" "An open-cover model for attached cells — 1 item · 1 fully proved" click n4 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#empty-cell-attachment-iso" "An attachment with no cells is an isomorphism — 1 item · 1 fully proved" click n5 "../../../../roadmap/fundamental-group/van-kampen/index.html#attaching-spheres-generate-intersection" "The attaching-sphere pieces generate the intersection group — 1 item · 1 fully proved" click n6 "../../../../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 n7 "../../../../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 n8 "../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#binary-cover-trivial-piece-quotient" "A binary cover with a trivial second fundamental group — 1 item · 1 fully proved" class n9 focus click n9 "../../../../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 n10 "../../../../roadmap/fundamental-group/van-kampen/presentation-complex/index.html#relator-cell-attachment" "Free-group relators form a standard two-cell attachment — 1 item · 1 fully proved" click n11 "../../../../roadmap/fundamental-group/van-kampen/presentation-complex/index.html#presentation-complex-fundamental-group" "The presentation complex has the presented fundamental group — 1 item · 1 fully proved" click n12 "../../../../roadmap/fundamental-group/van-kampen/presentation-complex/index.html#cyclic-presentation-complex" "The cyclic presentation complex — 1 item · 1 fully proved"