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A basic consequence of the Seifert–van Kampen theorem is the following.

Theorem: Consider a union of topological spaces $X$, $Y$ whose intersection $X\cap Y = Z$ is open connected and $\pi_1$-injective. Then for a basepoint $z_0 \in Z$ we have $$ \pi_1(X\cup Y, z_0) = \pi_1(X,z_0)*_{\pi_1(Z,z_0)}\pi_1(Y,z_0), $$ i.e. the fundamental group of the union is the amalgamated product.

I am looking for references for two variations of this result:

  1. What if we assume $X$, $Y$ are compact CW-complexes and $Z = X \cap Y$ is a $\pi_1$-injective subcomplex? The difference here is that $Z$ is no longer open but as a subcomplex it's still a sensible subspace.
  2. $X$ has two disjoint homeomorphic $\pi_1$-injective sensible subspaces $Z_1,Z_2 \subset X$ (e.g. points, circles) and the quotient space $Y = X/{\sim}$ is obtained by identifying $Z_1$, $Z_2$. Then $\pi_1(Y)$ is isomorphic to an HNN extension of $\pi_1(X)$.

The type of application I'm thinking about is stuff along the lines of identifying two boundary components of a surface with boundary or identifying two different points in a space and then working out the resulting fundamental group.

I'm not happy with what is in Munkres, Hatcher, or Scott and Wall, in that there are no clear results I can just give to a grad student (maybe that's the point though). Is anybody aware of other references that provide a good account of this or is it just folklore?

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    $\begingroup$ All these results follow from considering the $2$-skeleton of the associated CW complex and applying SVK. I don't think of them as folklore, but just a step away from the basic theory. $\endgroup$
    – Ryan Budney
    Commented Jun 25 at 21:35
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    $\begingroup$ For the CW complex variant, what you need is that a subcomplex $Y$ of a CW complex $X$ has an open neighborhood that deformation retracts to it. Hatcher calls such an $(X,Y)$ a “good pair”, and verifies this (if I recall correctly) in the appendix on the topology of CW complexes. $\endgroup$
    – Andy Putman
    Commented Jun 25 at 21:40
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    $\begingroup$ (your second conclusion is something I often give as an exercise, and if a grad student can’t prove it for themselves they don’t really understand SvK) $\endgroup$
    – Andy Putman
    Commented Jun 25 at 21:41
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    $\begingroup$ As Andy Putman said, the CW variant is in my book. It's in the paragraph following the proof of Proposition A.5 in the Appendix. The book should have included a cross reference to this in the discussion of van Kampen's theorem in Chapter 1, but I'm revising the book and will add a cross reference. $\endgroup$
    – Allen Hatcher
    Commented Jun 26 at 9:01
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    $\begingroup$ For the second variant, there is some discussion of HNN extensions in Example 1B.13 of my book, but this doesn't quite give what you're looking for. I'll try to include this somewhere in the revised edition. $\endgroup$
    – Allen Hatcher
    Commented Jun 26 at 9:11

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