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On the growth of the homology of a free loop space. (English) Zbl 1300.55016

A conjecture of Vigué-Poirrier asserts that, for a simply connected, finite, rationally hyperbolic CW-complex \(X\), the homology \(H_*(X^{S^1};\mathbb Q)\) of the free loops on \(X\) with rational coefficients grows exponentially. In this paper, the authors are able to prove that some classes of spaces satisfy a stronger condition which we now recall: let \(X\) be a simply connected space with rational homology of finite type and such that \(0<\,\)log index\(\,H_*(\Omega X;\mathbb Q)<\infty\). Then, \(X^{S^1}\) is said to have good exponential growth if it has “controlled exponential growth” (imposed by a technical condition) and log index\(\,H_*(\Omega X;\mathbb Q)=\,\)log index\(\,H_*(X^{S^1};\mathbb Q)\).
First, it is proved that, if \(X\) is a simply connected wedge of spheres of finite type then \(X^{S^1}\) has good exponential growth.
Also, given a fibration \(F\to E\to B\) of simply connected spaces with finite type rational homology, interesting consequences are obtained by comparing good growth and log index of the spaces in the fibration induced by taking free loops \(F^{S^1}\to E^{S^1}\to B^{S^1}\). Explicitly, if log index\(\,\pi_*(F)<\)log index\(\,\pi_*(E)\) and \(E^{S^1}\) has good exponential growth, then \(B^{S^1}\) has good exponential growth and log index\(\,H_*(B^{S^1};\mathbb Q)=\,\)log index\(\,H_*(E^{S^1};\mathbb Q)\). On the other hand, if log index\(\,\pi_*(B)<\,\)log index\(\,\pi_*(E)\), then \(F^{S^1}\) has good exponential growth if and only if \(E^{S^1}\) does and in this case, log index\(\,H_*(F^{S^1};\mathbb Q)=\,\)log index\(\,H_*(E^{S^1};\mathbb Q)\).
The paper ends with a selected list of carefully chosen examples to which all of this can be applied. For instance, the rational homology of the free loops on a configuration space of two particles in a closed \(2\)-connected manifold has good exponential growth, as long as the rational cohomology of the manifold is not generated by a single element, and its homotopy Lie algebra is finitely generated.
All of the above is achieved via an exhaustive and deep analysis of the log index and controlled exponential growth of Sullivan algebras and Sullivan extensions, in the context of rational homotopy theory.

MSC:

55P62 Rational homotopy theory
55P35 Loop spaces
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