Rational obstruction theory and rational homotopy sets

Abstract

We develop an obstruction theory for homotopy of homomorphisms f, g : M → N between minimal differential graded algebras. We assume that M = ΛV has an obstruction decomposition given by V = V 0 ⊕ V 1 and that f and g are homotopic on ΛV 0 . An obstruction is then obtained as a vector space homomorphism V 1 → H * (N ). We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes [M, N ]. This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A, B] when A and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps.Mathematics Subject Classification (1991):55P62, 55Q05, 55S35, 55P10

Realisability problem in arrow categories

“…, and therefore j 2 (g −1 2 ) = 0. Sog −1 2 ∈ π 2 (H) and k 2 (g −1 2 ) =g −1 2 , which from the previous equations, leads us to ϕ 0 (g −1 2 ]. Hence, by Lemma 2.5, we obtain that (g −1 1 ,g −1 2 ) ∈ H, and therefore (g 1 ,g 2 ) ∈ H.…”

“…Families of elliptic CDGA's. We follow a similar approach as in [4,5] where, to every finite simple graph G, a minimal Sullivan algebra M G is assigned in such a way that the group of self-homotopy equivalences of M G is isomorphic to the automorphism group of G. Our construction in [5, Definition 2.1] was based on a homotopically rigid algebra given in [2], and it was functorial only when restricted to the subcategory of full graph monomorphisms (see [5,Remark 2.8]). However, the morphism ϕ obtained in Theorem 1.2 (see also Corollary 3.7) is not a full monomorphism in general, so our previous construction is useless to answer Question 1.1 in Arr(CDGA).…”

Homotopically rigid Sullivan algebras and their applications

“…A decade after, Arkowitz and Lupton came across an example of a minimal Sullivan algebra (equivalently, a rational homotopy type of a space) M with E(M ) trivial, [3,Example 5.1]. This, let us say homotopically rigid algebra, M provides us with motivation and guidance: in our paper [5], we use M as a building block to construct minimal Sullivan algebras, [5,Definition 2.1], with the property that their group of self-homotopy equivalences are isomorphic to an arbitrary finite group G previously fixed, [5,Theorem 1.1].…”

Realization of 2-solvable nilpotent groups as groups of classes of homotopy self-equivalences

“…A basic problem consists to know if all nilpotent group G can appear as ε (X) [2,11,4,8,9]. Partial answers were supplied by M. Arkowitz and G. Lupton [2,4], S. Piccarreta [11] and D. Kahn [8,9].…”

“…Partial answers were supplied by M. Arkowitz and G. Lupton [2,4], S. Piccarreta [11] and D. Kahn [8,9].…”