Showing papers in "Boletin De La Sociedad Matematica Mexicana in 1992"

On the character variety of group representations of a 2-bridge link p/3 into PSL(2,C)

Abstract: Consider the group G of a classical knot or link in S3. It is natural to consider the representations of G into PSL(2,C). The set of conjugacy classes of nonabelian representations is a closed algebraic set called the character variety (of representations of G into PSL(2,C)). If G is the group of a 2-bridge knot or link, then a polynomial results by an earlier published theorem of the authors. This polynomial is related to the Morgan-Voyce polynomials Bn(z), which can be defined by the formulas pn(z)=Bn(z−2), where pn=zpn−1−pn−2, p0=1, p1=z, or (z1−10)n=(pnpn−1−pn−1−pn−2). In this paper the authors do many calculations for classes of 2-bridge knots or links.

Nilpotence in the steenrod algebra

Abstract: While all of the relations in the Steenrod algebra, A, can be deduced in principle from the Adem relations, in practice, it is extremely difficult to determine whether a given polynomial of elements in A is zero for all but the most elementary cases. In his original paper [Mi] Milnor states “It would be interesting to discover a complete set of relations between the given generators of A”. In particular Milnor shows that every positive dimensional element of A is nilpotent. Thus it would be desirable to find a simple closed form for nilpotence relations in A. Let x ∈ A. We say that x has nilpotence k, if x = 0 and xk−1 6= 0 (take x = 1). In this case we write Nil(x) = k. In this paper we investigate Nil(x) for several infinite families of Milnor basis elements of A at the prime 2. The paper is organized as follows. First, an infinite family of subalgebras and isomorphisms between them are constructed. The isomorphisms are used to produce infinite families of elements having the same nilpotence. Next, we compute strong upper and lower bounds for the nilpotence of Milnor basis elements in these subalgebras. Comparing these bounds and extending to the families produced via the isomorphisms shows that Sq(2(2 − 1) + 1) has nilpotence k + 2. Finally a strong lower bound for the nilpotence of P s t is computed for all s, t ∈ N. The main results are stated and discussed in Sections II and III. Detailed proofs are presented in Section IV.

Les espaces de bracelets, les complexes de Stasheff et le groupe de Thompson

Abstract: Etant donne un bracelet b = {h 1 ,: .. , hn+1} il existe une unique isometrie de 1HI telle que l'image de h1 est\"{zl Imz = 1} et telle que l'image de hn+l est tangente a O; alors h1 n hn+l = {i} et [-oo, O] est partage en n intervalles par les points a l'i_nfini des hi. L'espace de tels bracelets \"normalises\" s'appelle Bn. On veut le comprendre. On observe d' abord que Bn est une variete C , diffeomorphe a JRn-2 • Or, les points a l'infini des hi, 2 ::; i ::; n-1, varient librement et leur configuration dans 8IHI fixe le bracelet ( voir I' Appendice). Nous allons etudier la topologie d'un certain sous-espace de Bn. Soit b E Bn, et soit a(b) l'arc ferme \"horocercle par morceaux\" qui parcourt les hi entre les points d'intersection avec hi-l et hi+l (voir fig. 1, ou a(b) est plus fonce). Si l'arc a(b) est simple (sans points d'autointersection) on dit que b est propre. Le sous-espace de Bn (evidemment ouvert) des bracelets propres s'appelle B~. Nous ecrivons aussi !'adherence B: = B~ U B~, ou B~ est l'espace de bracelets tangents, i.e. tels que a(b) possede au moins un point d'autointersection, mais tels qu'aucune intersection n'est transverse ( voir fig. 2). Notre but est de detailler la structure des B:.

Phantom maps and rational equivalences , ii

Abstract: We call a space X a finite type domain if it is a pointed, connected CWcomplex whose integral homology groups are finitely generated in each degree; a pointed space Y will be referred to as a finite type target if each of its homotopy groups is finitely generated. Recently, Gray and McGibbon studied the universal phantom map out of a given space, [3]. For a finite type domain X, they showed that Ph(X, Y) = 0 for all spaces Y (not just those of finite type) if and only if ZX is a retract of VnIXn. They also proved a /?-local version of this that is somewhat sharper; it says that if X is the /^-localization of a nilpotent, finite type domain, then Ph (X, Y) = 0 for all spaces Y if and only if IX has the homotopy type of a bouquet of finite dimensional spaces. They noted, however, that there are some spaces, such as X = RP??, for which the universal phantom map X ?> VnZXn is essential and yet Ph(X, Y) = 0 for every finite type target Y (see also [4]). The problem of characterizing those domains, X, that satisfy this weaker condition was left open in [3]. It is solved here.