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Let $M$ denote a complex manifold of dimension $n$ and let $X\subset M$ denote an analytic hypersurface. Then it is a standard fact from several complex variables that around a given point $p\in X$ there are open subsets $V\subset X, W\subset \mathbb{C}^{n-1}$ and and a finite-sheeted covering $\pi: V\rightarrow W$ branched over an analytic set $A$. Frequently, $W$ is taken to be a polydisc (see, for example, Griffiths & Harris). Now set $V' = V\setminus \pi^{-1}(A)$ and $W' = W\setminus A$. If $V'$ is connected, then $\pi$ induces a genuine (holomorphic) covering $\pi':V'\rightarrow W'$.

For appropriately chosen basepoints $v_0, w_0$ in $V',W'$, respectively, covering space theory says that $\pi'$ induces an injection $\pi_1(V',v_0)\rightarrow \pi_1(W',w_0)$, and in some cases (e.g. involving curves) this might be enough to figure out $\pi_1(V',v_0)$ if $\pi_1(W',w_0)$ is known. If $V$ were smooth, $\pi^{-1}(A)$ would be an analytic subset of complex codimension 1 and the inclusion $V'\hookrightarrow V$ would induce a surjection $\pi_1(V',v_0)\rightarrow \pi_1(V,v_0)$, determining a presentation for $\pi_1(V,v_0)$. However, $V$ may not be smooth, and I am not sure in what generality this map is still a surjection, although clearly I'd like to know.

What I really want to do is determine a presentation for $\pi_1(X,x_0)$. It seems that one way to to go about doing this would be to construct an open cover $(V_{\alpha})$ of $X$ consisting of open sets with the same properties possessed by $V$ above AND such that $(V_{\alpha})$ satisfies the hypotheses of the Seifert-van Kampen Theorem (see, for example, Hatcher). In the very best case, one could arrange for the intersections $V_{\alpha}\cap V_{\beta}$ to be simply-connected and for the triple intersections to be path-connected. Then one could read off a presentation of $\pi_1(X,x_0)$ in terms of the presentations of the $\pi_1(V_{\alpha}, v_{0\alpha})$.

I'd be interested to know when such a cover $(V_{\alpha})$ exists, especially for $n\geq 3$.

I am also interested in hearing about any known methods of calculating the fundamental group of an analytic hypersurface (not necessarily smooth), in general or in special cases (for example, the Lefschetz Hyperplane Theorem can be used on certain projective hypersurfaces).

(Note: I'd like to consult Dimca's book "Singularities and Topology of Hypersurfaces", but I'll be away from my library for the next few weeks.)

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Probably you already know, but Lefschetz theorem for homotopy groups says that if $X$ is a non-singular irreducible complex projective variety of dimension $n$ and $D$ any effective ample divisor on $X$, then the homomorphisms $\pi_i(D)\to\pi_i(X)$ induced by inclusion are bijective for $i\le n-2$ and surjective if $i=n-1$. Note that the theorem does not assume that $D$ is non-singular! However, one cannot dispense entirely with the non-singularity of $X$... –  diverietti Jul 3 '12 at 8:28
Yes, I am familiar with Lefschetz for homotopy groups. Thank you. In fact, one can say a little more in case $X\subset \mathbb{P}^{n+1}$ is a hypersurface. The $d$-uple Veronese embedding $\nu_d: \mathbb{P}^{n+1}\rightarrow \mathbb{P}^N$ allows us to view $X$ itself as a hyperplane section of the image $\nu_d(\mathbb{P}^{n+1})$, which is $(n+1)$-dimensional. Then Lefschetz gives us $\pi_{n-1}(X) = \pi_{n-1}(\mathbb{P}^{n+1})$. As a result, a hypersurface in $\mathbb{P}^3$ is simply-connected. –  Kevin Jul 3 '12 at 14:53
Replay to Kevin's comment: I think that what you are saying is not more than what I was saying... It's just my statement applied to the case when the ambient space is the projective space. –  diverietti Jul 3 '12 at 15:03
About your original question. Sorry, I don't understand one thing: you are taking $X$ to be smooth, right? So couldn't you take as $V$ just a coordinate chart and $W\subset\mathbb C^{n−1}$ to be the image of this coordinate chart, so that it is simply a biholomorphism ? –  diverietti Jul 3 '12 at 15:04
You are absolutely right on both counts. Regarding my comment: I was simply saying that, using the result you mentioned, one can go a little further in calculating the homotopy groups of $X$, i.e. the case $i=n-1$. Regarding my original question: At first I wanted to allow for $X$ to be singular, but at some point changed my mind as I was posting the question. I'll edit the question to reflect your comment. –  Kevin Jul 3 '12 at 15:55
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1 Answer

up vote 3 down vote accepted

It should be said that van Kampen's paper "On the connection between the fundamental groups of some related spaces". Amer. J. Math. 55 (1933) 261--267, gives a formula for the case of a union of two spaces with non-connected intersection, and this was needed for his work on algebraic curves: "On the Fundamental Group of an Algebraic Curve". Amer. J. Math. 55 (1933), no. 1-4, 255–267. The non-connected case seems to me best handled in modern terms using groupoids. (See my web pages.) (An earlier version of the theorem in the connected case and for simplicial complexes was given by Seifert.)

Since homotopy groups are mentioned in the last comment, I mention the higher order theorems of the Seifert-van Kampen type of which the 2-d version was given in my paper with Philip Higgins , ``On the connection between the second relative homotopy groups of some related spaces'', Proc. London Math. Soc. (3) 36 (1978) 193-212. This uses extensively the notion of crossed module, and has been applied to give explicit calculations of homotopy 2-types, and second homotopy groups. (Again, see references on my web pages).

In view of the provenance of van Kampen's paper, it would be very interesting to know if such higher theorems are applicable to the situation of the question.

@Kevin Kordek: September 2013: I should add that Grothendieck was interested in these possibilities. See the last problem stated in the set of problems and "Future directions?" given in our new book on "Nonabelian algebraic topology"; more details and pdf's available from here. Part of his comment is as follows:

"It seems to me, in any case, that this $\underset{\to}{\lim}$-operation ["higher order van Kampen theorem"] in the context of homotopy types is of a very fundamental character, with wide range of theoretical applications. To give just one example, relying on the existence of such a formalism, it is possible to give a very simple explicit algebraic description of the full homotopy types of the Mumford-Deligne compactifications of the modular topoi for complex curves of given genus $g$, say, with $\nu$ "marked" points, in terms essential1y of such a (finite) direct limit of $K(\pi, 1)$-spaces, where $\pi$ ranges over certain "elementary" Teichm\"uller groups (those, roughly, corresponding to modular dimension $\leq 2$), and to give analogous descriptions, too, of all those subtopoi of the previous one, deducible from its canonical "stratification" at infinity by taking unions of strata. In fact, such descriptions should apply to any kind of ``stratified" space or topos, as it can be expressed (in an essentially canonical way, which apparently was never made explicit yet in this literature) as a (usually finite) direct limit of simpler spaces, namely the "strata", and "tubes" around strata, and "junctions" of tubes, etc."

So I probably missed out on not pursuing this, partly because of pursuing work with J.-L. Loday on even more powerful Higher van Kampen type theorems.

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The OP might like to know that the 'usual' way of dealing with the case of non-connected intersectionn is with HNN extensions; this is equivalent to the groupoid point of view. –  HJRW Jul 3 '12 at 11:50
@Ronnie Brown "In view of the provenance of van Kampen's paper, it would be very interesting to know if such higher theorems are applicable to the situation of the question." I completely agree. Thanks for answering, and for the references. –  Kevin Jul 3 '12 at 21:09
@HW Thanks. Good to know. –  Kevin Jul 3 '12 at 21:10
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