Paul-Eugène Parent

Emergence of the Witt group in the cellular lattice of rational spaces

Keywords: cellular space ; quadratic form ; Witt group ; Quillen ; model Reference GR-HE-ARTICLE-2002-003doi:10.1090/S0002-9947-02-03049-0View record in Web of Science Record created on 2008-11-14, modified on 2017-05-12

Simulations as Homotopies

We exhibit a model structure on 2-Cat, obtained by transfer from sSet across the adjunction C2 Sd 2 a Ex 2 N2. A certain class of homotopies in this model structure turns out to be in 1-to-1 correspondence with strong simulations among labeled transitions systems, formalising the geometric intuition of simulations as deformations. The correspondence still holds in the cubical setting, characterising simulations of higher-dimensional transition systems (HDTS).

Rational Lusternik–Schnirelmann category of fibrations

Abstract If F → E → B is a fibration, a classical result of Varadarajan asserts that cat E⩽ cat F+ cat B( cat F+1) , where cat S denotes the Lusternik–Schnirelmann category of S . We give improved upper bounds in the rational case of the form cat 0 E⩽ cat 0 F+ cat 0 B( cat 0 F+2−r 0 F), where r 0 F is a new invariant, namely the rational retraction index of F satisfying depth F⩽r 0 F⩽ cat 0 F, so that we recover the classical formula when r 0 F =1. However, the retraction index is often larger than 1, and in particular, we prove that if H ∗ (F; Q ) is a Poincare duality algebra with at least 2 generators, then r 0 F ⩾2, giving the bound of (Contemp. Math. 227 (1996) 177) without their dimension hypothesis. Moreover, if F is coformal, then r 0 F= cat 0 F , which yields the much lower estimate cat 0 E⩽ cat 0 F+2 cat 0 B.

Density and unique decomposition theorems for the lattice of cellular classes

A classC of pointed spaces is called a cellular class if it is closed under weak equivalences, arbitrary wedges and pointed homotopy pushouts. The smallest cellular class containingX is denoted byC(X), and a partial order relation ≪ is defined by:X ≪Y ifY εC(X). In this text we investigate the sub partial order sets generated respectively by simply connected finite CW-complexes and by rational spaces. For rational spaces we prove a unique decomposition theorem, a density theorem and the existence of infinitely many non-comparable elements. We then prove the density theorem for a generic class of finite CW-complexes.

Using the topological characterization of synchronous models

Abstract This paper contributes to the characterization of synchronous models of distributed computing using topological techniques. We consider a generic synchronous model with send-omission failures and use a topological structure corresponding to a bounded number of rounds of the model. We observe some nice properties of the structure and derive from these properties necessary and sufficient conditions to solve consensus in this model.

LS category: product formulas

Given two continuous maps f:X --> Y and g:W --> Z between simply connected CW-complexes of finite type, we show that Mcat(k) (f x g) = Mcat(k)f + Mcat(k)g over any field k. Moreover, we establish a characterization Mcat(k)f = e(C*(Y;k))C(*)(X; k), where e is a kind of Toomer invariant introduced by Felix et al. (1998). Finally, we apply the characterization to show that Mcat(Q)f = Mcat(o)f

Homotopy localization functor L-f with respect to maps f having a wedge of spheres as homotopy cofibre

In the connected case, we compute explicity thef-localization (in the sense of [3]) for the class of mapsZ(n)↪Z in which the cofibre is a wedge of spheres. We have an analogous result over the rationals where the cofibre is arbitrary.