A RATIONAL TORSION INVARIANT

JOHN EWING, PETER LÖFFLER, AND ERIK KJÆR PEDERSEN

Abstract. 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

K_{1}(Qπ_{1}(X)/Σ) ⊆ K_{1}(Qπ_{1}(X))

where Σ = ∑_{g∈π_{1}(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 K_{1}(Qπ_{1}(X)) = K_{1}(Qπ_{1}(X)/Σ) × K_{1}(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 K_{1}(Q)/K_{1}(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^{1} 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 over Q, Λ = Λ[x_{1}, x_{2}, …, x_{n}] where deg x_{i} 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 K_{1}(Q).

For the proof we need a lemma.

An algebra automorphism is given by its values on the generators x_{i}, We say an automorphism is elementary if it is the identity on all but one of the x_{i}, and the one exceptional x_{i} is mapped to x_{i} + monomial, where monomial is not a multiple of x_{i}.

Lemma 1.2. The torsion of an elementary automorphism is trivial.

Proof. Suppose the elementary automorphism λ is the identity except on x_{i} and let deg(x_{i}) = 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(x_{i}) = x_{i} + m, where m is the monomial. In degrees higher than k, write Λ_{j} = V ⊕ W, where W is generated as a vector space by monomials that are divisible by x_{i}, 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 x_{i} or m times the monomial is zero. On W any monomial μ is sent to μ+(μ/x_{i})·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(x_{i}) = α_{1}X_{i}) 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 x_{i}. Suppose x_{i} is not sent to a multiple of itself, and all x_{i} in smaller degrees are. Then x_{i} must be sent to

∑ s_{k}x_{k} + ∑ t_{j}m_{j},

where s_{k} and t_{j} are rational numbers and the m_{j} are monomials, which are products of elements of lower degree. Composing with the elementary automorphisms that send x_{i} to x_{i} − t_{j}m_{j}, we see that x_{i} is now sent to ∑ s_{k}x_{k}. 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. Let X 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 K_{1}(Q)/K_{1}(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) ∈ K_{1}(Q) = Q^{∗}. A different choice of integral basis will vary q(X) by an element in the image of therefore well defined in K_{1}(Q)/K_{1}(Z) = Q_{+}^{∗}.

In case X is also nilpotent with finite fundamental group π, there is another quite geometric way to describe the invariant q(X). We describe this below.

Suppose H^{∗}(X; Q) = Λ[x_{1}, …, x_{n}] with degree(x_{i}) = 2k_{i} + 1. By obstruction theory we obtain a homotopy equivalence

X_{(0)} → ∏_{i=1}^{n} K(Q, 2k_{i} + 1) = ∏_{i=1}^{n} S_{(0)}^{2k_{i}+1}.

Also by obstruction theory, applied to the diagram

\xymatrix{ & X \ar[d] \\ ∏_{i=1}^{n} S^{2k_{i}+1} \ar[r] & X_{(0)}}

one finds a map ∏_{i=1}^{n} S^{2k_{i}+1} → X which induces a rational homotopy equivalence. (To see this, note that any obstruction will have finite order and by composition with a rational equivalence of the product of spheres can be made to vanish.)

The map ∏_{i=1}^{n} S^{2k_{i}+1} → X now induces a map of chain complexes

Λ[x_{1}, x_{2}, …, x_{n}] → S_{∗},

which is a homotopy equivalence after tensoring with Q. The basis in the homology of ∏ S^{2k_{i}+1} is exactly sent to a basis which is dual of the monomial basis in S_{∗} ⊗ Q. Hence, the invariant q(X) can be interpreted as the torsion of this rational homotopy equivalence.

This naturally leads to the multiplicative Euler characteristic, yet another way to define q(X).

Definition 2.3. Let f : X → Y be a rational homotopy equivalence of spaces of finite type. The multiplicative Euler characteristic of f is defined to be

mχ(f) = ∏ \frac{|H_{odd}(f)|}{|H_{even}(f)|}.

Lemma 2.4. Let f : X → Y be a rational homotopy equivalence of finitely dominated spaces. Let γ ∈ Q_{+}^{∗} denote the morion of the homotopy equivalence. Then γ = mχ(f).

Proof. Let S_{∗} and T_{∗} be finitely generated free chain complexes representing the singular chains of X and Y respectively. Let f_{∗} : S_{∗} → T_{∗} be a chain map representing f. To compute the torsion of f, one chooses bases for S_{∗} and T_{∗}, and then notes that the mapping cone of f_{∗} is acyclic after tensoring with Q. But the homology of f is by definition the homology of the mapping cone.

Such an integral chain complex is the sum of chain complexes of type

0 → Z ⟶^{n} Z → 0,

where n ≠ 0, since the chain complex is Q-acyclic. But the torsion of this short chain complex is simply n or 1/n (depending on the parity). This is, however, exactly the order of the homology □

Remark 2.5. The interpretation of the multiplicative Euler characteristic as the torsion of the rational homotopy equivalence makes certain properties self-evident. For example, for the composition of rational homotopy equivalences

X ⟶^{f} Y ⟶^{g} Z

it is now clear that mχ(g · f) = mχ(g) · mχ(f).

3. Obstruction to S^{1} actions. Suppose X is a nilpotent complex of finite type with finite fundamental group, and suppose the rational cohomology of X is an exterior algebra on odd dimensional generators. The invariant q(X) is then defined and we have shown in the preceding section that it can be interpreted as the torsion (or multiplicative Euler characteristic) of a rational homotopy equivalence from a product of spheres to X. We shall now show that q(X) is an obstruction to a free S^{1} action on the homotopy type of X having quotient space finitely dominated.

Theorem 3.6. Let X be as above. Suppose there is a fibration S^{1} → X → Y, orientable, with Y of finite type. Then q(X) = 1 ∈ Q_{+}^{∗}.

Proof. It is easy to check that the space Y is nilpotent. Consider the rationalization of the fibration S_{(0)}^{1} → X_{(0)} → Y_{(0)}. A spectral sequence argument shows that

Y_{(0)} = CP_{(0)}^{n} × ∏ S_{(0)}^{2k_{i}+1}.

Using obstruction theory (as in §2), we produce a map

CP^{n} × ∏ S^{2k_{i}+1} → Y

which is a rational homotopy equivalence. The S^{1}-fibration S^{1} → X → Y is classified by BSG(2) = BSO(2) = CP^{∞}, and so is given by a 2-dimensional cohomology class. Pulled back to CP^{n} × ∏ S^{2k_{i}+1}, this must be the 2-dimensional generator of H^{2}(CP^{n}).

Hence we obtain a pullback diagram:

[figure: S^{1}; S^{1}; S^{2n−1} × ∏ S^{2k_{i}+1}; g; S^{2n−1}/(Z/k) × ∏ S^{2k_{i}+1}; f; X; CP^{n} × ∏ S^{2k_{i}+1}; h; Y]

Now from §2 we know that q(X) = mχ(f · g) = mχ(f) · mχ(g). It is easy to see that mχ(g) = 1, so we need only compute mχ(f).

The Serre spectral sequence to compute H_{∗}(f; Z) has

E_{p,q}^{2} = H_{p}(S^{1}; H_{q}(h)).

Thinking of E_{∗,∗}^{2} as singly graded by total degree, it is clear that mχ(E_{∗,∗}^{2}) = 1, since E_{0,∗}^{2} = E_{1,∗}^{2} and E_{s,∗}^{2} = 0 for s ≠ 0, 1. Also, by a standard argument, the Euler characteristic mχ does not change by taking homology. Hence,

mχ(E_{∗,∗}^{2}) = mχ(E_{∗,∗}^{3}) = mχ(E_{∗,∗}^{∞}).

But E_{∗,∗}^{∞} is the associated bigraded object for the Serre filtering of the graded module H_{∗}(f). Hence mχ(H_{∗}(f)) = mχ(E_{∗,∗}^{∞}) = 1. □

This of course implies q(G) = 1 for any Lie group. It is an interesting question whether this is true for any H-space. Notice this may also be asked p-locally.

4. Linking numbers. In one special situation the invariant q(X) turns out to coincide with a well-known classical invariant.

Let S_{1}^{n}, S_{2}^{n}, and S_{3}^{n+k+1} be simply connected oriented manifolds having the rational homotopy type of spheres.

Suppose we have imbeddings

S_{1}^{n} ⟶_{i_{1}} S_{3}^{n+k+1} ⟵_{i_{2}} S_{2}^{k}

such that i_{1}(S_{1}) ∩ i_{2}(S_{2}) = ∅.

Now consider the following diagram:

[figure: 0; H_{k+1}(S_{3}, S_{3} − S_{1}); ∂; H_{k}(S_{3} − S_{1}); H_{k}(S_{3}); Z · [S_{1}^{n}] ≅ H_{n}(S_{1}); H_{k}(S_{2}); i_{2}^{∗}; ≅; Z · [S_{2}^{k}]]

We have H_{k}(S_{3}) ⊕ Q = 0. Hence there are integers r, s ∈ Z such that

s · i_{2} · [S_{2}^{k}] = r · ∂[S_{1}^{n}].

Define the linking number l(S_{3} : S_{1}, S_{2}) of S_{1} and S_{2} in S_{3} to be

l(S_{3}; S_{1}, S_{2}) = r/s.

This generalizes the usual notion of linking numbers of spheres in spheres. We say that S_{1} and S_{2} are non-trivially linked in S_{3} if l(S_{3}; S_{1}, S_{2}) ≠ 0. For each of the above manifolds we have a degree one map α_{i} : S_{i}^{s} → S^{s}. Let

M(S_{i}^{s}) = m_{x}(α_{i}) = \frac{∏ |TorsH_{odd}(S_{i})|}{∏ |TorsH_{even}(S_{i})|}.

If S_{1} and S_{2} are nontrivially linked in S_{3}, then it is easy to check that the inclusion S_{1}^{n} → S_{3}^{n+k+1} − S_{2}^{k} is a rational homotopy equivalence and hence X = S_{3} − S_{1} − S_{2} is rationally S^{n} × S^{k}.

Proposition 4.7. Let S_{1}^{n}, S_{2}^{k} ⊆ S_{3}^{n+k+1} be rational homology spheres in S. Let X = S^{n+k+1} − S_{1}^{n} − S_{2}^{k}. If n and k are odd we have

q(x) = \frac{M(S_{3})}{M(S_{1}) · M(S_{2})} · l(S_{3}; S_{1}, S_{2}).

In particular if all rational homotopy spheres are genuine spheres our invariant is just the classical linking number.

Proof. Let r_{1} ↓ s_{1} be the normal bundle to S_{1} in S_{3}. Consider the inclusion j_{1} : Sr_{1} → X. A straightforward calculation shows

m_{x}(j_{1}) = \frac{M(S_{3})}{M(S_{1}) · M(S_{2})} · l(S_{3}; S_{1}, S_{2}).

Let j̃_{2} : S^{n} → S_{1}^{n} be a map of nonzero degree (such a map exists since S_{1}^{n} is a rational homology sphere). This induces

j_{2} : j̃_{2}^{∗}(Sr_{1}) → Sr_{1}.

A spectral sequence argument as in §3 shows m_{x}(j_{2}) = 1. In analogy to [1] we see that Sr_{1} ↓ S_{1} is rationally fiber homotopically trivial. Hence there is a map

\xymatrix{S^{n} × S^{k} \ar[d] \ar[r]_{j_{3}} & j_{2}^{∗}(Sr_{1}) \ar[d] \\ S^{n} \ar[r]_{j̃_{3}} & S^{n}}

induced by a nonzero degree map j̃_{3}. As before we have m_{x}(j_{3}) = 1.

From §2 we know

q(x) = m_{x}(j_{3} · j_{2} · j_{1}) = m_{x}(j_{1}) · 1 · 1

= \frac{M(S_{3})}{(M(S_{1}) · M(S_{2})} · l(S_{3}; S_{1}, S_{2}).

□

5. *. References

(1959), 959–964.

1. W. Massey, On the normal bundle of a sphere imbedded in euclidean space, Proc. Amer. Math. Soc. 10 2. G. Mislin, Finitely dominated nilpotent spaces, Ann. of Math. (2) 103 (1976), 547–556.

A RATIONAL TORSION INVARIANT 7

Department of Mathematics, Indiana University, Bloomington, Indiana 47405 Mathematisches Institut der Universität, 3400 Göttingen, West Germany Mathematical Institute, Odense University, Odense DK5000, Denmark