Van Kampen’s theorem result

There is a more general version of the theorem of van Kampen which involves the fundamental groupoidMathworldPlanetmathPlanetmathPlanetmath π1(X,A) on a set A of base points, defined as the full subgroupoid of π1(X) on the set AX. This allows one to compute the fundamental groupMathworldPlanetmathPlanetmath of the circle S1 and many more cases.

Theorem If X is the union of open sets U,V with intersection W, and A meets each path component of U,V,W then the following induced diagram


is a pushout in the category of groupoidsPlanetmathPlanetmath.

This may be found in R. Brown’s book “Topology: a geometric account of general topology and the fundamental groupoid”, Ellis Horwood 1988 (first edition McGraw Hill, 1968). It has many useful applications, and was the guide for higher dimensional theorems involving higher homotopy groupoids.

Title Van Kampen’s theorem result
Canonical name VanKampensTheoremResult
Date of creation 2013-03-22 14:09:23
Last modified on 2013-03-22 14:09:23
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 6
Author mathcam (2727)
Entry type Result
Classification msc 55Q05