Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Hopf Invariants in Real and Rational Homotopy Theory
Stockholm University, Faculty of Science, Department of Mathematics. (Topology)ORCID iD: 0000-0001-6941-3979
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.
Keywords [en]
Rational homotopy theory, Real homotopy theory, operads, Hopf invariants
National Category
Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-146246ISBN: 978-91-7649-980-1 (print)ISBN: 978-91-7649-981-8 (electronic)OAI: oai:DiVA.org:su-146246DiVA, id: diva2:1136442
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At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript.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
List of papers
1. Algebraic Hopf Invariants
Open this publication in new window or tab >>Algebraic Hopf Invariants
(English)Manuscript (preprint) (Other academic)
Keywords
Hopf invariants, operads, rational homotopy theory
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-146243 (URN)
Available from: 2017-08-28 Created: 2017-08-28 Last updated: 2017-09-07Bibliographically approved
2. Hopf invariants and differential forms
Open this publication in new window or tab >>Hopf invariants and differential forms
(English)Manuscript (preprint) (Other academic)
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-146245 (URN)
Available from: 2017-08-28 Created: 2017-08-28 Last updated: 2017-09-07Bibliographically approved
3. HOMOTOPY MORPHISMS BETWEEN CONVOLUTION HOMOTOPY LIE ALGEBRAS
Open this publication in new window or tab >>HOMOTOPY MORPHISMS BETWEEN CONVOLUTION HOMOTOPY LIE ALGEBRAS
(English)Manuscript (preprint) (Other academic)
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-146244 (URN)
Available from: 2017-08-28 Created: 2017-08-28 Last updated: 2017-09-07Bibliographically approved

Open Access in DiVA

fulltext(233 kB)67 downloads
File information
File name FULLTEXT02.pdfFile size 233 kBChecksum SHA-512
9710e63ced9dfc4b6847529feb840a49bda159a11310e10b6a9eca295b97a721619284b1c88bf6be956f957559cc54a29a82928b017b8b2d4bb3d82d817ba8fa
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Wierstra, Felix
By organisation
Department of Mathematics
Geometry

Search outside of DiVA

GoogleGoogle Scholar
Total: 67 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
Total: 279 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf