An attachment with no cells is an isomorphism: 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. fully proved6Proved, and every prerequisite is fully proved too.
graph LR
n0("An indexed family of cones has a two-set open cover"):::fully_proved
n1["Every abstract cell attachment has an indexed cone model"]:::fully_proved
n2["An attachment with no cells is an isomorphism"]:::fully_proved
n3["Attaching higher cells preserves the fundamental group"]:::fully_proved
n4["Attaching 2-cells adds the attaching relations"]:::fully_proved
n5("Higher-cell attachments preserve path components"):::fully_proved
n1 --> n2
n1 -.-> n5
n2 -.-> n3
n2 --> n4
n2 -.-> n5
n0 --> n2
n0 -.-> n5
click n0 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#indexed-cone-open-cover" "An indexed family of cones has a two-set open cover — 1 item · 1 fully proved"
click n1 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#abstract-cell-attachment-indexed-model" "Every abstract cell attachment has an indexed cone model — 1 item · 1 fully proved"
class n2 focus
click n2 "../../../../../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 n3 "../../../../../roadmap/fundamental-group/van-kampen/index.html#attach-higher-cells-fundamental-group" "Attaching higher cells preserves the fundamental group — 1 item · 1 fully proved"
click n4 "../../../../../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 n5 "../../../../../roadmap/fundamental-group/van-kampen/cell-attachment-support/index.html#higher-cell-attachment-path-components" "Higher-cell attachments preserve path components — 1 item · 1 fully proved"