34
$\begingroup$

The theorem of Seifert-Van-Kampen states that the fundamental group $\pi_1$ commutes with certain colimits. There is a beautiful and conceptual proof in Peter May's "A Concise Course in Algebraic Topology", stating the Theorem first for groupoids and then gives a formal argument how to deduce the result for $\pi_1$. However, there we need the assumption that our space is covered by path-connected open subsets, whose finite intersections are path-connected again. This assumption can be weakened using a little "perturbation" trick which is explained in the proof given by Hatcher in his book "Algebraic Topology". After doing this, we only need that triple intersections are path connected (and we cannot do better).

Question 1. Is it possible to "conceptualize" the perturbation trick, thus weakening the assumption concerning finite intersections? Maybe this is answered by one of the pure categorical proofs of Seifert-van-Kampen?

Question 2. In practice (explicit calculations of fundamental groups), do we actually often need or use that this weakening of the assumption is possible?

  • 3
    since you haven't received any responses at MATH.SE, I suggest you to pose this question at MO2011-07-03

1 Answers 1