Talk:Vanishing cycle

Disambiguation

In fact, there are two different notions of what is called "vanishing cycle", one from the singular bundles (the one that is described now in the article) and another from the foliations theory (the one used by Sullivan in his proof of Novikov's theorem, in the paper "Cycles for the dynamical study of foliated manifolds and complex manifolds", Invent. Math., 36 (1976), p. 225-255.). The latter ones are those that are nontrivial on a given leaf of a foliation, but become (homologically or homotopically) trivial if being displaced on an arbitrarily small leaf. This leads to important conclusions: the discs they bound form a "Folner" sequence, and hence the presence of such cycles of codimension one implies the existence of transverse invariant measure.

So: maybe it worth either making a disambiguation, or mentioning directly both notions in the same article? --Burivykh (talk) 13:32, 13 May 2010 (UTC)[reply]

P.S.: A google books prooflink: [1]. --Burivykh (talk) 07:27, 19 May 2010 (UTC)[reply]

Todo

This page should include a discussion of the basics behind nearby and vanishing cycles including both the arithmetic and geometric formulations. The geometric side should demonstrate how the pullback along the covering map adds the monodromy information with an informative example. On the arithmetic side there should be a discussion about semi-stable arithmetic surfaces (these have double point singularities). Also, this page should include information about what a Henselian trait is and how this relates to arithmetic surfaces (including a mention of arithmetic deformations). — Preceding unsigned comment added by Username6330 (talk • contribs) 01:15, 11 November 2017 (UTC)[reply]

Also some physicist should come here and talk about axions. --Fourier-Deligne Transgirl (talk) 08:29, 11 March 2023 (UTC)[reply]