A Rational Torsion Invariant

We show that for spaces with rational cohomology an exterior algebra on odd dimensional generators, one can define a torsion invariant which is a rational number. This may be interpreted as an absolute version of the multiplicative Euler characteristic associated to a rational homotopy equivalence. 0. Introduction. Reidemeister torsion of a finite complex X, with trivial action of the fundamental group on rational homology, takes values in K1(Qπ1(X)/Σ) ⊆ K1(Qπ1(X)) where Σ = ∑ g∈π1(X) g. It is the purpose of this paper to show that one can sometimes define a “Reidemeister torsion” invariant in the other factor of K1(Qπ1(X)) = K1(Qπ1(X)/Σ) × K1(Q). Specifically, when a space with finitely generated integral homology has rational cohomology an exterior algebra on odd dimensional generators, then we show that there is a torsion invariant defined in K1(Q)/K1(Z) = Q ∗ +. This may be interpreted as an absolute version of the multiplicative Euler characteristic of a rational homotopy equivalence that has been utilized by a number of authors (J. Davis, P. Löffler, G. Mislin, and S. Weinberger). In case the space X is also nilpotent with finite fundamental group, we show that this invariant is an obstruction to a free S action on the homotopy type of X having finitely dominated quotient. We also show that the invariant can be used to extend the definition of linking number of odd dimensional spheres to rational homology spheres. 1. Algebraic preliminaries. Consider an exterior algebra overQ, Λ = Λ[x1, x2, . . . , xn] where deg xi is odd. We are interested in the torsions of algebraic isomorphisms of Λ. Theorem 1.1. If n > 1 then any algebra automorphism of Λ has trivial torsion in K1(Q). For the proof we need a lemma. An algebra automorphism is given by its values on the generators xi, We say an automorphism is elementary if it is the identity on all but one of the xi, and the one exceptional xi is mapped to xi + monomial, where monomial is not a multiple of xi. Lemma 1.2. The torsion of an elementary automorphism is trivial. 1 2 JOHN EWING, PETER LÖFFLER, AND ERIK KJÆR PEDERSEN Proof. Suppose the elementary automorphism λ is the identity except on xi and let deg(xi) = k. In degrees lower than k, λ is the identity. In degree k, λ is a a usual elementary automorphims of a vector space. Now suppose λ(xi) = xi +m, where m is the monomial. In degrees higher than k, write Λj = V ⊕W , where W is generated as a vector space by monomials that are divisible by xi, but do not have a common factor with the monomial m, and V is generated by all other monomials. The automorphism λ is the identity on V , since for each monomial either the monomial does involve xi or m times the monomial is zero. On W any monomial μ is sent to μ+(μ/xi) ·m, showing that λ is a product of usual elementary automorphisms of a vector space. Hence, in each degree, λ is elementary. It follows that the torsion of λ is trivial. Proof of Theorem 1.1. Let Φ be an algebra automorphism of Λ. If Φ is a Hopf algebra automorphism (that is Φ(xi) = α1Xi) the result is easy to show, and has been exploited by Mislin [2, p.555]. We shall now convert Φ to a Hopf algebra automorphism by composition with elementary automorphisms. In the lowest dimension we can clearly change Φ to be diagonal by composition with elementary automorphisms. We then proceed by induction on the degree of the xi. Suppose xi is not sent to a multiple of itself, and all xi in smaller degrees are. Then xi must be sent to ∑ skxk + ∑ tjmj, where sk and tj are rational numbers and the mj are monomials, which are products of elements of lower degree. Composing with the elementary automorphisms that send xi to xi − tjmj, we see that xi is now sent to ∑ skxk. This however, is the usual linear situation which can always be diagonalized by composition with elementary automorphisms. Since elementary automorphisms have trivial torsion by Lemma 1.2, we conclude that Φ itself has trivial torsion. 2. Definition of the invariant. LetX be a space of finite type (that is,H∗(X) is finitely generated) and suppose H∗(X;Q) is an exterior algebra on odd dimensional generators. We can now define the invariant in K1(Q)/K1(Z) = Q ∗ + mentioned in the introduction. We begin by choosing a finitely generated free chain complex S∗ which is homotopy equivalent to the singular chains of X. In H∗(S∗;Q) = H∗(X;Q), choose a basis dual to the basis for H(S∗;Q) given by the monomials. By Theorem 1.1, this gives a well-defined basis for H∗(S∗;Q) up to simple isomorphism (when measured in Q ∗). Now in S∗ we may choose an integral basis which then induces a basis for S∗ ⊗Q. Since the chain complex is based, and the homology is based as well, we obtain as usual a torsion element q(X) ∈ K1(Q) = Q∗. A different choice of integral basis will vary q(X) by an element in the image of therefore well defined in K1(Q)/K1(Z) = Q ∗ +. In caseX is also nilpotent with finite fundamental group π, there is another quite geometric way to describe the invariant q(X). We describe this below. A RATIONAL TORSION INVARIANT 3 Suppose H∗(X;Q) = Λ[x1, . . . , xn] with degree(xi) = 2ki + 1. By obstruction theory we obtain a homotopy equivalence