Van Kampen’s theorem

The proof of Van Kampen s theorem is elementary, though it requires some topological finesse. We refer the reader to any of the standard textbooks. 

Proof. These are both corollaries of Van Kampen s Theorem just substitute the assumptions in diagram (6.3) and take the colimit. ... 

Example 6.26. Let us calculate the fundamental group of the projective plane RP using Van Kampen s theorem. Consider a standard representation of KP as a unit disk in the plane with the boundary self-identified by the antipodal map, i.e., x —x. Let A be the disk centered at the origin with radius 1/2, and let B be obtained from MP by removal of the interior of A. We have Au B = RP, and it is easy to equip RP with a CW structure such that A and B are CW subcomplexes (alternatively, one could thicken A and B a little bit so as to obtain a topologically identical open covering of RP ). 

In either case. Van Kampen s theorem can be applied. The space A is contractible, the intersection A fl B is homeomorphic to a circle, and the space B deformation retracts to a circle. To see the last of these facts, simply retract the punctured unit disk to its boundary, and note that the antipodal self-identification of the boundary again produces a circle. We have nfiA) = 0 and 7Ti(B) = Tri(AnB) = Z. The fundamental group homomorphisms induced by the inclusion maps i AC B A and An B B are the following ... 

This same equation has been independently derived by Caceres [90] using van Kampen s lemma [91] and the Bourret-Frisch-Pouquet theorem [92], while the theory adopted by Annunziato et al. [87] rests essentially on the Zwanzig approach of Section III, namely, a Liouville-like perspective. 

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