DVI File: freebks.dvi
Title: Volumes, middle-dimensional systoles, and Whitehead products
Authors: Ivan K. Babenko, Mikhail G. Katz and Alexander I. Suciu
Subj-class: 53C23 (Primary) 55Q15 (Secondary)
Comments: LaTeX2e, 10 pages. Also available at dg-ga/9707016
Let (X,g) be a closed, orientable Riemannian manifold of
dimension 2m. The k-systole of (X,g), sys_k(g), is the infimum
of areas of non-bounding cycles represented by maps of k-dimensional
manifolds into X. We are interested in the following question:
Does there exist a constant, C, such that every metric g on X satisfies
(*) sys_{m}^{2}(g) <= C vol_{2m}(g) ?
If there is no such C, we say that X is ``systolically free."
In the case of surfaces of positive genus the answer to (*) is affirmative.
In the case m >= 2, this question has been referred to by M. Gromov
as the ``basic systolic problem." Progress on the problem became
possible once Gromov described a special family of metrics on
S^1 x S^3 and surgical procedures suitable for generalizations.
In this note, we prove that closed manifolds of dimension 2m >= 6
with torsion-free middle-dimensional homology are systolically free.
An underlying theme of this paper is the influence of homotopy theory
on the geometric inequality (*). Our basic topological tools are the
Hilton-Milnor theorem and theorems of Eckmann and G. Whitehead on
composition maps in homotopy groups of spheres. Our geometric tools
are the coarea inequality and pullback arguments for simplicial metrics.