Hostname: page-component-6898cfcb55-zsp5v Total loading time: 0 Render date: 2026-11-11T14:18:46.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinite dimensional cycles associated to operators

Published online by Cambridge University Press:  22 January 2016

Hiroshi Morimoto
Hiroshi Morimoto*
Affiliation:
Department of Mathematics, School of Science, Nagoya University, Nagoya 464-01, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

A family of operators defined on infinite dimensional spaces gives rise to interesting cycles (or subvarieties) of infinite dimension which represent a topological or non-topological feature of operator families. In this paper we will give a general theory of these cycles, and give some estimates among them. We will apply this theory, in the final section, to cycles derived from Dirac operators.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 127 , September 1992, pp. 1 - 14
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1992

References

[ 1 ] Atiyah, M.F. Drinfeld, V.G. Hitchen, NJ. Manin, Yu.L, Costruction of instantons, Phys. Lett, 65 A (1978), 185–187.CrossRefGoogle Scholar
[ 2 ] Atiyah, M.F. Jones, -J.D.S., Topological aspects of Yang-Mills theory, Comm. Math. Phys., 61 (1978), 97–118.CrossRefGoogle Scholar
[ 3 ] Boardman, J.M., Singularities of differentiable maps, I.H.E.S. Publ. Math., 33 (1967), 21–57.Google Scholar
[ 4 ] Douady, A., Un espace de Banach dont le groupe linéaire n’est pas connexe, Indag. Math., 68(1965), 787–789.Google Scholar
[ 5 ] Elworthy, K.D. Tromba, -A., Differential structures and Fredholm maps on Banach manifolds, Proc. Sympos. Pure Math., 15 (1970), 95–113.Google Scholar
[ 6 ] Gromov, M. Lawson, -H.B., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, I.H.E.S. Publ. Math., 58 (1983), 83–196.Google Scholar
[ 7 ] Koschorke, U., Infinite dimensional K-theory and characteristic classes of Fredholm bundle maps, Proc. Sympos. Pure Math., 15 (1970), 95–113.CrossRefGoogle Scholar
[ 8 ] Kuiper, N., The homotopy type of the unitary group of Hilbert space, Topology, 3 (1965), 19–30.Google Scholar
[ 9 ] Lichnerowicz, A., Spineurs harmoniques, C. R. Acad. Sci. Paris, Ser. A-B, 257 (1963), 7–9.Google Scholar
[10] Morimoto, H., On certain complex analytic cobordism between subvarieties realizing Chern classes of bundles, Nagoya Math. J., 88 (1982), 121–132.Google Scholar
[11] Morimoto, H., On the generic existence of holomorphic sections and complex analytic bordism, Proc. Japan Acad., 58 Ser. A, No. 7 (1982), 306–308.Google Scholar
[12] Morimoto, H., Realization of Chern classes by subvarieties with certain singularities, Nagoya Math. J., 80 (1980), 49–74.Google Scholar
[13] Morimoto, H., Some Morse theoretic aspects of holomorphic vector bundles, Advanced Studies in Pure Math., 3, Geometry of Geodesies and related Topics, North-Holland, (1984), 283–389.Google Scholar
[14] Morimoto, H., A theory of infinite dimensional cycles for Dirac operators, Proc. Japan Acad., 65, Ser. A, No. 3(1989), 67–69.Google Scholar
[15] Morimoto, H., A Yang-Mills gradient flow and its infinite dimensional limit cycles, Berkeley MSRI (1989), 03924–89.Google Scholar
[16] Ronga, F., Les class duales aux singularités d’ ordre deux, Comment. Math. Helv., 47 (1972), 15–35.Google Scholar
[17] Thom, R., Structural stability and morphogenesis, Bonjamin, New York, 1972.Google Scholar
[18] Zeeman, E. C., Catastrophe Theory, selected papers, 1972–1977, Addison-Wesley Publ. Comp. Inc., 1977.Google Scholar