Stephen Halperin

Obstructions to homotopy equivalences

Abstract An obstruction theory is developed to decide when an isomorphism of rational cohomology can be realized by a rational homotopy equivalence (either between rationally nilpotent spaces, or between commutative graded differential algebras). This is used to show that a cohomology isomorphism can be so realized whenever it can be realized over some field extension (a result obtained independently by Sullivan). In particular an algorithmic method is given to decide when a c.g.d.a. has the same homotopy type as its cohomology (the c.g.d.a. is called formal in this case). The chief technique is the construction of a canonically filtered model for a commutative graded differential algebra (over a field of characteristic zero) by perturbing the minimal model for the cohomology algebra. This filtered model is also used to give a simple construction of the Eilenberg-Moore spectral sequence arising from the bar construction. An example is given of a c.g.d.a. whose Eilenberg-Moore sequence collapses, yet which is not formal.

Finiteness in the minimal models of Sullivan

Let X be a 1-connected topological space such that the vector spaces I7I*(X) 0 Q and H*(X; Q) are finite dimensional. Then H*(X; Q) satisfies Poincare duality. Set Xr, = E(I)Pdim rlp(X) 0 Q and X, = X(I)Pdim HP(X; Q). Then Xri 0. Moreover the conditions: (1) Xrn = 0, (2) X, > 0, H*(X; Q) evenly graded, are equivalent. In this case H*(X; Q) is a polynomial algebra truncated by a Borel ideal. Finally, if X is a finite 1-connected C.W. complex, and an r-torus acts continuously on X with only finite isotropy, then Xn < r.

Adams Cobar Equivalence

Let F be the homotopy fibre of a continuous map Y --> omega X, with X simply connected. We modify and extend a construction of Adams to obtain equivalences of DGAs and DGA modules, OMEGA-C*(X) --> congruent-to CU*(OMEGA-X), and OMEGA(C*omega(Y); C*(X)) --> congruent-to CU*(F), where on the left-hand side OMEGA(-) denotes the cobar construction. Our equivalences are natural in X and omega. Using this result we show how to read off the algebra H*(OMEGA-X; R) and the H*(OMEGA-X; R) module, H*(F; R), from free models for the singular cochain algebras CS*(X) and CS*(Y); here we assume R is a principal ideal domain and X and Y are of finite R type.

Universal enveloping algebras and loop space homology

Abstract Let R be a principal ideal domain containing 1 2 , and let ρ(R) be the least non-invertible prime in R. We prove the following theorem: Let X be an r-connected CW-complex (r≥1) such that dim X≤rρ(R). If H∗(ΩXiR) is torsion free, then there is a graded Lie algebra E and a natural isomorphism UE → ≌ H ∗ (ΩX;R) of graded Hopf algebras.

The Homotopy Lie-algebra for Finite Complexes

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Formal Spaces With Finite-dimensional Rational Homotopy

Let S be a simply connected space. There is a certain principal fibration K1 E -Ko in which K1 and Ko are products of rational Eilenberg-Mac Lane spaces and a continuous map 4: S E such that in particular 00 = 4T o 4 maps the primitive rational homology of S isomorphically to that of Ko. A main result of this paper is the THEOREM. If dim sro(S) 0 Q < oo then 4) is a rational homotopy equivalence if and only if all the primitive homology in H*(S; Q) and H*(KO, S; Q) can (up to integral multiples) be represented by spheres and disk-sphere pairs. COROLLARY. If S is formal, 4 is a rational homotopy equivalence.

Deviations from the linear mixture rule in nonequilibrium chemical kinetics

In experimental studies of vibrational relaxation, dissociation, or isomerization of molecular gases, it is common to use mixtures of the reacting gas with inert diluents. Rate constants for the reaction in pure reactant gas or pure diluent gas are then evaluated by extrapolation using the linear mixture rule (LMR): kLMR=∑ixiki, where xi is the mole fraction of gas i and ki is the rate constant for the reaction in pure component i. However, this rule is only obeyed rigorously for first order or pseudo‐first‐order processes having a single rate‐determining step or which proceed at thermal equilibrium. We demonstrate theoretically that deviations from the linear mixture rule are always positive or zero (ktrue≥kLMR) and show with model calculations that the neglect of these deviations resulting from the use of the linear mixture rule may lead to large overestimates of the true pure component rate constants.

Engel Elements in the Homotopy Lie-algebra

An Engel element in a graded Lie algebra, L, (over a field k) is an element x ϵ L such that ad x is locally nilpotent. The linear span of these elements is a graded subspace E ⊂L. The depth of L is the infinum (possibly ∞) of the set of integers m for which ExtULm(k,UL)≠0. It is known that if L is the rational homotopy Lie algebra of a simply connected space, X, or the homotopy Lie algebra of a local noetherian ring, A, then the depth of L is bounded above respectively by the Lusternik-Schnirelmann category of X and the embedding dimension of A. Theorem. If L is concentrated in degrees >0 (or in degrees <0) and if depth L = m then there are at most m linearly independent Engel elements of even degree: ∑idimkE2i⩽depthL

The Category of a Map and the Grade of a Module

AbstractLetf: Y → X be a continuous map between connectedCW complexes. The homologyH*(F) of the homotopy fibre is then a module over the loop space homologyH* (ΩX). Theorem:If H*(F; R) and H*(ΩX; R) are R-free (R a principal ideal domain) then for some H*(ΩX; R)-projective moduleP=P>0and for some m ≤ cat f: