Twisted stable homotopy theory
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.
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| Format: | Thesis |
| Language: | eng |
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Massachusetts Institute of Technology
2006
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| Online Access: | http://hdl.handle.net/1721.1/33095 |
| _version_ | 1811097631140610048 |
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| author | Douglas, Christopher L |
| author2 | Michael J. Hopkins. |
| author_facet | Michael J. Hopkins. Douglas, Christopher L |
| author_sort | Douglas, Christopher L |
| collection | MIT |
| description | Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. |
| first_indexed | 2024-09-23T17:02:32Z |
| format | Thesis |
| id | mit-1721.1/33095 |
| institution | Massachusetts Institute of Technology |
| language | eng |
| last_indexed | 2024-09-23T17:02:32Z |
| publishDate | 2006 |
| publisher | Massachusetts Institute of Technology |
| record_format | dspace |
| spelling | mit-1721.1/330952019-04-12T09:40:17Z Twisted stable homotopy theory Douglas, Christopher L Michael J. Hopkins. Massachusetts Institute of Technology. Dept. of Mathematics. Massachusetts Institute of Technology. Dept. of Mathematics. Mathematics. Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005. Includes bibliographical references (p. 133-137). There are two natural interpretations of a twist of stable homotopy theory. The first interpretation of a twist is as a nontrivial bundle whose fibre is the stable homotopy category. This kind of radical global twist forms the basis for twisted parametrized stable homotopy theory, which is introduced and explored in Part I of this thesis. The second interpretation of a twist is as a nontrivial bundle whose fibre is a particular element in the stable homotopy category. This milder notion of twisting leads to twisted generalized homology and cohomology and is central to the well established field of parametrized stable homotopy theory. Part II of this thesis concerns a computational problem in parametrized stable homotopy, namely the determination of the twisted K-homology of the simple Lie groups. In more detail, the contents of the two parts of the thesis are as follows. Part I: I describe a general framework for twisted forms of parametrized stable homotopy theory. An ordinary parametrized spectrum over a space X is a map from X into the category Spec of spectra; in other words, it is a section of the trivial Spec- bundle over X. A twisted parametrized spectrum over X is a section of an arbitrary bundle whose fibre is the category of spectra. I present various ways of characterizing and classifying these twisted parametrized spectra in terms of invertible sheaves and local systems of categories of spectra. I then define homotopy-theoretic invariants of twisted parametrized spectra and describe a spectral sequence for computing these invariants. (cont.) In a more geometric vein, I show how a polarized infinite-dimensional manifold gives rise to a twisted form of parametrized stable homotopy, and I discuss how this association should be realized explicitly in terms of semi-infinitely indexed spectra. This connection with polarized manifolds provides a foundation for applications of twisted parametrized stable homotopy to problems in symplectic Floer and Seiberg-Witten-Floer homotopy theory. Part II: I prove that the twisted K-homology of a simply connected simple Lie group G of rank n is an exterior algebra on n - 1 generators tensor a cyclic group. I give a detailed description of the order of this cyclic group in terms of the dimensions of irreducible representations of G and show that the congruences determining this cyclic order lift along the twisted index map to relations in the twisted ... bordism group of G. by Christopher Lee Douglas. Ph.D. 2006-06-19T17:39:58Z 2006-06-19T17:39:58Z 2005 2005 Thesis http://hdl.handle.net/1721.1/33095 62174705 eng M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582 137 p. 7023296 bytes 7030385 bytes application/pdf application/pdf application/pdf Massachusetts Institute of Technology |
| spellingShingle | Mathematics. Douglas, Christopher L Twisted stable homotopy theory |
| title | Twisted stable homotopy theory |
| title_full | Twisted stable homotopy theory |
| title_fullStr | Twisted stable homotopy theory |
| title_full_unstemmed | Twisted stable homotopy theory |
| title_short | Twisted stable homotopy theory |
| title_sort | twisted stable homotopy theory |
| topic | Mathematics. |
| url | http://hdl.handle.net/1721.1/33095 |
| work_keys_str_mv | AT douglaschristopherl twistedstablehomotopytheory |