The group presentation associated to an open cover: 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 proved7Proved, and every prerequisite is fully proved too.
graph LR
n0["The group presentation associated to an open cover"]:::fully_proved
n1("The van Kampen cover map is surjective"):::fully_proved
n2("Overlap relations vanish under the cover map"):::fully_proved
n3["Cover factorizations of a loop"]:::fully_proved
n4("Elementary factorization moves preserve the quotient class"):::fully_proved
n5("Homotopic factorizations have the same quotient class"):::fully_proved
n6("The kernel is generated by overlap relations"):::fully_proved
n3 --> n4
n3 --> n5
n0 --> n3
n0 --> n2
n0 --> n4
n0 --> n5
n0 --> n6
n0 --> n1
n2 -.-> n6
n4 -.-> n5
n5 -.-> n6
class n0 focus
click n0 "../../../../roadmap/fundamental-group/van-kampen/index.html#cover-group-presentation" "The group presentation associated to an open cover — 1 item · 1 fully proved"
click n1 "../../../../roadmap/fundamental-group/van-kampen/index.html#van-kampen-surjective" "The van Kampen cover map is surjective — 1 item · 1 fully proved"
click n2 "../../../../roadmap/fundamental-group/van-kampen/index.html#cover-map-relations" "Overlap relations vanish under the cover map — 1 item · 1 fully proved"
click n3 "../../../../roadmap/fundamental-group/van-kampen/index.html#cover-factorization" "Cover factorizations of a loop — 1 item · 1 fully proved"
click n4 "../../../../roadmap/fundamental-group/van-kampen/index.html#factorization-moves" "Elementary factorization moves preserve the quotient class — 1 item · 1 fully proved"
click n5 "../../../../roadmap/fundamental-group/van-kampen/index.html#homotopic-factorizations" "Homotopic factorizations have the same quotient class — 1 item · 1 fully proved"
click n6 "../../../../roadmap/fundamental-group/van-kampen/index.html#van-kampen-kernel" "The kernel is generated by overlap relations — 1 item · 1 fully proved"