I've seen many times the following application of the SVK theorem:

Let $M$ and $N$ two smooth $n$-manifolds ($n\ge 3$) with boundary and suppose that they have the same boundary $B$. Now, after glueing $M$ and $N$ along $B$ we obtain: $$X=M\cup_{B} N$$ At this point we can apply the SVK to the triple $M$, $N$, $M\cap N=B$ in order to calculate the fundamental group of $X$.

The problem is that $M$, $N$ and $B$ are in general not open in $X$ (once that they are glued together). On the other hand we know that the openness condition is necessary in the proof of SVK theorem.

  • 3
    $\begingroup$ While the openness condition in SvK's theorem is essential (and there are counterexamples otherwise), but unless you're in a pathological case, it's often possible to find a slightly bigger open set that deformation retracts onto the subspace you want, just like in Michael Albanese's answer. $\endgroup$ Jan 8 '16 at 9:35

Let $M$ be a smooth manifold with boundary. A neighborhood of $\partial M$ is called a collar neighbourhood if it is the image of a smooth embedding $[0, 1)\times\partial M \to M$ that restricts to the identification $\{0\}\times\partial M \to \partial M$.

Every smooth manifold with nonempty boundary has a collar neighbourhood - see Lee's Introduction to Smooth Manifolds (second edition), Theorem $9.25$.

As $B$ is the boundary of $M$, it has a collar neighbourhood $C \subset M$ and as $B$ is the boundary of $N$, it has a collar neighbourhood $D \subset N$. Then $C\cup D$ becomes an open neighbourhood of $B$ in $X = M\cup_B N$. Furthermore, $C\cup D$ deformation retracts onto $B$, so $\pi_1(C\cup D, b_0) \cong \pi_1(B, b_0)$ for all $b_0 \in B$.

So we really want to apply the Seifert-van Kampen Theorem to the following open sets of $X$: $M\cup D$ (which deformation retracts to $M$), $C\cup N$ (which deformation retracts to $N$), and $C\cup D$ (which deformation retracts to $B$).

Note, it doesn't matter which collar neighbourhoods we take as the fundamental groups of the resulting open sets are independent of $C$ and $D$.

  • $\begingroup$ Wait a moment, now $C\cup D$ is open but $M$ and $N$ are not open. Maybe you want to apply SVK to $M\cup D$, $N\cup C$, $C\cup D$ $\endgroup$
    – manifold
    Jan 8 '16 at 1:37
  • $\begingroup$ Yes, I just realised I was being a bit sloppy. What you suggest should be fine. $\endgroup$ Jan 8 '16 at 1:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.