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Elimination of extremal index zeroes from generic paths of closed \(1\)-forms

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An Erratum to this article was published on 08 October 2014

Abstract

Let \( \alpha \) be a Morse closed \( 1 \)-form of a smooth \( n \)-dimensional manifold \( M \). The zeroes of \( \alpha \) of index \( 0 \) or \( n \) are called centers. It is known that every non-vanishing de Rham cohomology class \( u \) contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path \( (\alpha _t)_{ t\in [0,1] }\) of closed \( 1 \)-forms in a fixed class \( u\ne 0 \) such that \( \alpha _0,\alpha _1 \) have no centers, can be modified relatively to its extremities to another such path \( (\beta _t)_{t \in [0,1]} \) having no center at all.

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Notes

  1. In the mentioned paper, Novikov called “semi–open” his homology.

  2. Remark that for any \(s\in [0,2\varepsilon ]\), the function \(F_s^{i,\varepsilon }\) has no critical point near \(\partial {\fancyscript{C}}'_{b,s}\).

  3. Holes appear when the unstable manifold crosses a level containing a critical point of \(h_t\) in its closure.

  4. The other case says that \(k'gC_t'=k'gC_t\) is already under \(gC_t\).

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Acknowledgments

I am indebted by the continuous support I have received from François Laudenbach as well as for sharing with me his valuable point of view of mathematics. I am strongly grateful to Jean-Claude Sikorav for the comments that he made on a previous version of the paper, which have substantially improved the author’s understanding about the contributions that have been done to Novikov theory, as well as the accuracy of some citations on the present paper. I greatly appreciate the comments from the anonymous referee, specially for having pointed out an accident that was not treated in the first version of the paper. The first version was partially written at the University of Nantes under the financial support of the “ERC GEODYCON project” (eOTP: 12GEODYC). The revisions and improvements which have led to this final version were made during a “JSPS Postdoctoral Fellowship” under remarkably good research conditions at The University of Tokyo.

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  1. Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo, 153-8914, Japan

    Carlos Moraga Ferrándiz

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Moraga Ferrándiz, C. Elimination of extremal index zeroes from generic paths of closed \(1\)-forms. Math. Z. 278, 743–765 (2014). https://doi.org/10.1007/s00209-014-1332-4

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