Variations on Van Kampen’s Method
نویسنده
چکیده
We give a detailed account of the classical Van Kampen method for computing presentations of fundamental groups of complements of complex algebraic curves, and of a variant of this method, working with arbitrary projections (even with vertical asymptotes). Introduction In the 1930’s, Van Kampen described a general technique for computing presentations of fundamental groups of complements of complex algebraic curves. Though Van Kampen’s original approach was essentially valid, some technical details were not entirely clear and were later reformulated in more modern and rigorous terms (see for example the account by Chéniot, [C]). It is possible to transform Van Kampen’s “method” into an entirely constructive algorithm. To my knowledge, two implementations have been realized, one by Jorge Carmona, the other by Jean Michel and myself (GAP package VKCURVE, [VK]). The goal of the present note is to clarify some aspects which are usually neglected but must be addressed to obtain an efficient implementation. Also, the “Van Kampen’s method” explained here differs from the classical one, which assumes the choice of a “generic” projection: our variant method works with an arbitrary projection. The reason for what may appear to be a superfluous sophistication (since “generic” projections always exists and are easy to find) is that working with a non-generic projection may be computationally more efficient. The variant method explained here is implemented in VKCURVE, and has already been used to find previously unknown presentations. Let P ∈ C[X, Y ]. The equation P (X, Y ) = 0 defines an algebraic curve C ∈ C. Our goal is to find a presentation for the fundamental group of C − C (the method can be adapted to work with projective curves, as it is briefly mentioned at the end of section 2). Without loss of generality, we may (and will) assume that P is quadratfrei. View P as a polynomial in X depending on the parameter Y : P = α0(Y )X d + α1(Y )X d−1 + · · ·+ αd(Y ), with α0(Y ) 6= 0. To study C − C, we decompose it according to the fibers of the projection C → C, (x, y) 7→ y. Up to changing the variables, one could assume that d equals the total degree of P (the projection is then said to be “generic”); however, for reasons detailed
منابع مشابه
On the Gaussian approximation for master equations
We analyze the Gaussian approximation as a method to obtain the first and second moments of a stochastic process described by a master equation. We justify the use of this approximation with ideas coming from van Kampen’s expansion approach (the fact that the probability distribution is Gaussian at first order). We analyze the scaling of the error with a large parameter of the system and compar...
متن کاملThe Fundamental Group and Seifert-van Kampen’s Theorem
The fundamental group is an essential tool for studying a topological space since it provides us with information about the basic shape of the space. In this paper, we will introduce the notion of free products and free groups in order to understand Seifert-van Kampen’s Theorem, which will prove to be a useful tool in computing fundamental groups.
متن کاملFundamental Groups and the Van Kampen’s Theorem
In this paper, we define the fundamental group of a topological space and explore its structure, and we proceed to prove Van-Kampen’s Theorem, a powerful result useful for calculating the fundamental groups of spaces which decompose into spaces whose fundamental groups are already known. With these tools, we show that the circle, 2-sphere, torus, and figure-8 space are topologically distinct. W...
متن کاملHigher Minors and Van Kampen’s Obstruction
We generalize the notion of graph minors to all (finite) simplicial complexes. For every two simplicial complexes H and K and every nonnegative integer m, we prove that if H is a minor of K then the non vanishing of Van Kampen’s obstruction in dimension m (a characteristic class indicating non embeddability in the (m−1)-sphere) for H implies its non vanishing for K. As a corollary, based on res...
متن کاملThe Fundamental Group
The focus of this exposition is the fundamental group of a topological space. The paper begins by defining paths and homotopy, and proceeds to construct the fundamental group. Later, attention turns to the consequences of the construction, in particular to the fundamental group of S1 and Van Kampen’s Theorem.
متن کامل