This is a note on Mikhail Katz’s paper (1995) in which he constructed a sequence of Riemannian metrics on s.t. for . Where denotes the -systole which is the infimum of volumes of -dimensional integer cycles representing non-trivial homology classes. To find out more about systoles, here’s a nice 60-second introduction by Katz.
We are interested in whether there is a uniform lower bound for for being equipped with any Riemann metric. For , it is known that . Hence the construction gave counterexamples for all . An counterexample for is constructed later using different techniques.
The construction breaks into three parts:
1) Construction a sequence of metrics on s.t. approaches as .
2) Choose an appropriate metric on s.t. equipped with the product metric satisfy the property
3) By surgery on to obtain a sequence of metrics on , denote the resulting manifolds by , having the property that
The first two parts are done in previous notes (which are not published on this blog). Here I will talk about how is part 3) done given that we have constructed manifolds as in part 2).
Let equipped with metric as constructed in 1), be as constructed in 2).
Standard surgery: Let and let . . The resulting manifold from standard surgery along in is defined to be which is homeomorphic to .
We perform the standard surgery on the component of , denote the resulting manifold by . Hence equipped with some metric.
Note that the metric depends on the surgery and so far we have only specified the surgery in the topological sense. Now we are going to construct the surgery taking the metric into account.
First we pick to be a small ball of radius , call it . Pick that fills to be a cylinder of length for some large with a cap on the top. i.e. and . Hence the standard surgery can be performed with and . The resulting manifold is homeomorphic to and has a metric on it that depends on and .
Let i.e. the part that’s glued in during the surgery, call it the ‘handle’.
The following properties hold:
i) For any fixed , for sufficiently small,
implies can be made small by taking small.
ii) The projection of to its factor is distance-decreasing.
iii) If we remove the the cap part from (infact from ), then the remaining part admits a distance-decreasing retraction to .
i.e. project the long cylinder onto its base on which is .
iv) Both ii) and iii) remain true if we fill in the last component of i.e. replace it with and get a -dimensional polyhedron .
Since all we did in ii) and iii) is to project along the first and third component simultaneously or to project only the first component, filling in the third component does not effect the distance decreasing in both cases.
We wish to choose an appropriate sequence of and so that .
In the next part we first fix any and so that property i) from above holds and write for .
We are first going to bound all cycles with a nonzero component and then consider the special case when the cycle is some power of and this will cover all possible non-trivial cycles.
Claim 1: -cycle belonging to a class with nonzero -component, we have .
Note that since and by part 2), and by property i), . Let , hence . Therefore the bound in claim 1 would imply which is what we wanted.
a) If does not intersect
In this case the cycle can be “pushed off” the handle to lie in without increasing the volume. i.e. we apply the retraction from proposition iii).
b) If then by proposition ii), projects to its $S^n$ component by a distance-decreasing map and by construction in part 2).
Now suppose with .
Define s.t. .
Let , then by the coarea inequality, we have s.t. .
By our results in Gromov and the previous paper of Larry Guth or Wenger’s paper, s.t. -cycle with , . Hence with . By picking , we have as .
Recall that ; by construction and .
(1) If the cycle has non-trivial homology in , then by proposition iv), the analog of proposition iii) for implies we may retract to without decreasing its volume. Then apply case a) to the cycle after retraction we obtain .
Contradicting the assumption that .
(2) If has trivial homology in , then is a cycle with volume smaller than that’s contained entirely in . By case b), projects to its factor by a distance decreasing map, and . As above, , contradiction.