van Kampen's Theorem

In the usual diagram of inclusion homomorphisms, if the upper two maps are injective, then so are the other two.

More formally, consider a space  which is expressible as the union of pathwise-connected open sets , each containing the basepoint  such that each intersection  is pathwise-connected. Then, the homomorphism induced by the inclusion map from the free product of the fundamental groups of the s to the fundamental group of , i.e.,

(1)

is surjective (Hatcher 2001, p. 43). In addition, if each intersection  is pathwise-connected, then the kernel of  is the normal subgroup  generated by all elements of the form

(2)

where  is the homomorphism induced by the inclusion , and so  induces an isomorphism

(3)

REFERENCES:

Dodson, C. T. J. and Parker, P. E. A User's Guide to Algebraic Topology. Dordrecht, Netherlands: Kluwer, p. 88, 1997.

Hatcher, A. Algebraic Topology. Cambridge, England: Cambridge University Press, 2001.

Rolfsen, D. Knots and Links. Wilmington, DE: Publish or Perish Press, pp. 74-75 and 369-373, 1976.