Systolic topology

Given below is an attempt to put together a small synopsis of the history of the application of homotopy techiques in systolic geometry. Input from readers will be appreciated.

1. Such a history may start with D. Epstein's work on the degree of a map in the 1960's, continue with A. Wright's work on monotone mappings in the 1970's, then go on to developments in real semi-algebraic geometry which indicated that an arbitrary map can be homotoped to have good algebraic structure (see M. Coste), in the 1980's.

2. M. Gromov in his 1983 Filling paper in JDG goes out of the category of manifolds in order to prove the main isoperimetric inequality relating the volume of a manifold, to its filling volume. Namely, the cutting and pasting constructions in the proof of the main isoperimetric inequality involve objects more general than manifolds.

3. In the 1992 paper in Izvestia by I. Babenko , Lemma 8.4 is perhaps the place where a specific homotopy theoretic technique was first applied to systoles. Namely, this technique derives systolically interesting consequences from the existence of maps from manifolds to simplicial complexes. This work shows how the triangulation of a map f, based upon the earlier results mentioned in item 1 above, can help answer systolic questions, such as proving a converse to Gromov's central result of 1983. What is involved, roughly, is the possibility of pulling back metrics by f, once the map has been deformed to be sufficiently nice (in particular, real semialgebraic).

4. In 1992-1993, Gromov realized that a suitable oblique Z action on S3 × R gives a counterexample to a (1,3)-systolic inequality on S1 × S3. This example was described in Berger's survey in 1993. Berger also sketched Gromov's ideas toward constructing further examples of systolic freedom.

In the 1995 paper in Geom. Dedicata by M. Katz , metric simplicial complexes are used to prove the systolic freedom of the manifold Sn × Sn. In this paper, a polyhedron P is defined in equation (3.1). It is exploited in an essential way in an argument in the last paragraph on page 202, in the proof of Proposition 3.3. For an overview of systolic freedom, see the 2003 survey by Croke and Katz (msn).

5. The 2007 paper E_7, Wirtinger inequalities, Cayley 4-form, and homotopy exploits a map of classifying spaces BS1 → BS3 so as to construct a map from the quaternionic projective space to the complex projective space with cells attached in dimensions 3 (mod 4). It also connects (in a suitable sense) the quaternionic projective space to a hypothetical Joyce manifold (with Spin7 holonomy) with b4=1, relying upon a result by H. Shiga in rational homotopy theory.

6. An interesting related axiomatisation (in the case of 1-systoles) is proposed by M. Brunnbauer, who proves that the optimal systolic constant only depends on the image of the fundamental class in the classifying space of the fundamental group, generalizing earlier results of I. Babenko .

7. Another direction is the connection with a classical branch of topology, namely the Lusternik-Schnirelmann (LS) category. The connection can be summarized as follows. We define an invariant called SYSTOLIC CATEGORY, by analogy with the LS category. The invariant is defined roughly as follows. We consider universal volume lower bounds for an n-manifold manifold M. The type of lower bounds we consider are given by products of k-systoles of various dimensions k, up to a universal constant (independent of the metric). Thus, if the volume can be bounded below by an n-fold product of 1-systoles alone, we declare systolic category to be equal to n. Otherwise, systolic category will be strictly smaller than n. In the general case, systolic category is defined to be the maximal length of a product of systoles which can serve as such a universal volume lower bound. The details may be found in LS category and systolic category, etc. The invariant called systolic category turns out to coincide with the LS category for 2-complexes, for 3-manifolds, for simply connected 4-manifolds, as well as for all essential n-manifolds, when the common value of the two categories equals n. The latter property results from a combination of Gromov's theorem (see item 2 above) and I. Babenko's converse to it (see item 3 above). It can therefore be said that the LS category is sensitive to the systolic geometry of the underlying space.

8. The non-vanishing of C. Lescop's invariant (a generalisation of the Casson-Walker invariant) is a sufficient condition for the existence of suitable optimal systolic inequalities, as explained in our 2005 article. This direction was developed further by T. Cochran and J. Masters, as well as by A. Dranishnikov, M. Katz, and Y. Rudyak

Additional historical remarks may be found in the entries under Berger and Loewner.

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