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