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# Van Kampen’s theorem result

There is a more general version of the theorem of van Kampen which involves the fundamental groupoid $\pi_{1}(X,A)$ on a set $A$ of base points, defined as the full subgroupoid of $\pi_{1}(X)$ on the set $A\cap X$. This allows one to compute the fundamental group of the circle $S^{1}$ 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

$\xymatrix{\pi_{1}(W,A)\ar[r]\ar[d]&\pi_{1}(U,A)\ar[d]\\ \pi_{1}(V,A)\ar[r]&\pi_{1}(X,A)}$ |

is a pushout in the category of groupoids.

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.

## Mathematics Subject Classification

55Q05*no label found*

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## Comments

## van Kampen's theorem

I would be grateful if the reference to R. Brown's book â€œTopology: a geometric account of general topology and the fundamental groupoidâ€?, Ellis Horwood 1988 could also mention the latest edition `Topology and Groupoids' Booksurge PLC, 2006, which is available on amazon.com

Ronnie Brown