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Hopf invariants and differential forms
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0001-6941-3979
(English)Manuscript (preprint) (Other academic)
National Category
Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-146245OAI: oai:DiVA.org:su-146245DiVA: diva2:1136436
Available from: 2017-08-28 Created: 2017-08-28 Last updated: 2017-09-07Bibliographically approved
In thesis
1. Hopf Invariants in Real and Rational Homotopy Theory
Open this publication in new window or tab >>Hopf Invariants in Real and Rational Homotopy Theory
2017 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we use the theory of algebraic operads to define a complete invariant of real and rational homotopy classes of maps of topological spaces and manifolds. More precisely let f,g : M -> N be two smooth maps between manifolds M and N. To construct the invariant, we define a homotopy Lie structure on the space of linear maps between the homology of M and the homotopy groups of N, and a map mc from the set of based maps from M to N, to the set of Maurer-Cartan elements in the convolution algebra between the homology and homotopy. Then we show that the maps f and g are real (rational) homotopic if and only if mc(f) is gauge equivalent to mc(g), in this homotopy Lie convolution algebra. In the last part we show that in the real case, the map mc can be computed by integrating certain differential forms over certain subspaces of M. We also give a method to determine in certain cases, if the Maurer-Cartan elements mc(f) and mc(g) are gauge equivalent or not.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2017
Keyword
Rational homotopy theory, Real homotopy theory, operads, Hopf invariants
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-146246 (URN)978-91-7649-980-1 (ISBN)978-91-7649-981-8 (ISBN)
Public defence
2017-10-27, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 2: Manuscript. Paper 3: Manuscript.

Available from: 2017-10-04 Created: 2017-08-28 Last updated: 2017-09-20Bibliographically approved

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Citation style
  • apa
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  • Other style
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  • de-DE
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