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 X which is expressible as the union of pathwise-connected open sets A_alpha, each containing the basepoint x_0 in X such that each intersection A_alpha intersection A_beta is pathwise-connected. Then, the homomorphism induced by the inclusion map from the free product of the fundamental groups of the A_alphas to the fundamental group of X, i.e.,


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


where i_(alphabeta):pi_1(A_alpha intersection A_beta)->pi_1(A_alpha) is the homomorphism induced by the inclusion A_alpha intersection A_beta↪A_alpha, and so Phi induces an isomorphism

 pi_1(X) approx *_alphapi_1(A_alpha)/N.

See also

Knot Group

Portions of this entry contributed by Vidit Nanda

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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.

Referenced on Wolfram|Alpha

van Kampen's Theorem

Cite this as:

Nanda, Vidit and Weisstein, Eric W. "van Kampen's Theorem." From MathWorld--A Wolfram Web Resource.

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