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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 28th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave created an entry for _[[Bott periodicity]]_

   have created an entry for Bott periodicity

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeAug 29th 2013
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI haven't thought about this for quite a long time, but I once had the following "dream". Might it be possible to prove the Bott periodicity (say complex Bott periodicity, to keep things simple in the beginning), say in the form $K(X \times S^2) \cong K(X) \otimes K(S^2) = K(X)[\eta]/(\eta - 1)^2$, by calculating $K(S^2) \cong K(pt)[\eta]/(\eta - 1)^2$ constructively and interpreting that calculation in the topos of sheaves over $X$?

   I haven’t thought about this for quite a long time, but I once had the following “dream”. Might it be possible to prove the Bott periodicity (say complex Bott periodicity, to keep things simple in the beginning), say in the form $K(X \times S^2) \cong K(X) \otimes K(S^2) = K(X)[\eta]/(\eta - 1)^2$, by calculating $K(S^2) \cong K(pt)[\eta]/(\eta - 1)^2$ constructively and interpreting that calculation in the topos of sheaves over $X$?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 3rd 2020
   • (edited Nov 3rd 2020)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Max Karoubi]], _Bott Periodicity in Topological, Algebraic and Hermitian K-Theory_, In: [[Eric Friedlander]], [[Daniel Grayson]] (eds.) _[[Handbook of K-Theory]]_, Springer 2005 ([doi:10.1007/978-3-540-27855-9_4](https://doi.org/10.1007/978-3-540-27855-9_4)) <a href="https://ncatlab.org/nlab/revision/diff/Bott+periodicity/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/Bott+periodicity/9">v9</a>, <a href="https://ncatlab.org/nlab/show/Bott+periodicity">current</a>

   added pointer to

   • Max Karoubi, Bott Periodicity in Topological, Algebraic and Hermitian K-Theory, In: Eric Friedlander, Daniel Grayson (eds.) Handbook of K-Theory, Springer 2005 (doi:10.1007/978-3-540-27855-9_4)

   diff, v9, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeNov 9th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Michael Atiyah]], _Bott periodicity and the index of elliptic operators_, The Quarterly Journal of Mathematics, Volume 19, Issue 1, 1968, Pages 113–140 ([doi:10.1093/qmath/19.1.113](https://doi.org/10.1093/qmath/19.1.113)) <a href="https://ncatlab.org/nlab/revision/diff/Bott+periodicity/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/Bott+periodicity/10">v10</a>, <a href="https://ncatlab.org/nlab/show/Bott+periodicity">current</a>

   added pointer to:

   • Michael Atiyah, Bott periodicity and the index of elliptic operators, The Quarterly Journal of Mathematics, Volume 19, Issue 1, 1968, Pages 113–140 (doi:10.1093/qmath/19.1.113)

   diff, v10, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeNov 10th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Graeme Segal]], Prop. 3.2 in: _Equivariant K-theory_, Inst. Hautes Etudes Sci. Publ. Math. No. 34 (1968) p. 129-151 ([numdam:PMIHES_1968__34__129_0](http://www.numdam.org/item/PMIHES_1968__34__129_0)) <a href="https://ncatlab.org/nlab/revision/diff/Bott+periodicity/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/Bott+periodicity/11">v11</a>, <a href="https://ncatlab.org/nlab/show/Bott+periodicity">current</a>

   added pointer to:

   • Graeme Segal, Prop. 3.2 in: Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math. No. 34 (1968) p. 129-151 (numdam:PMIHES_1968__34__129_0)

   diff, v11, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 15th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin Schottenloher]], Section 15 of: _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Springer Lecture Notes in Physics __726__, 2008, ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf), [doi:10.1007/978-3-540-74956-1](https://link.springer.com/book/10.1007/978-3-540-74956-1)) <a href="https://ncatlab.org/nlab/revision/diff/Bott+periodicity/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/Bott+periodicity/13">v13</a>, <a href="https://ncatlab.org/nlab/show/Bott+periodicity">current</a>

   added pointer to:

   • Dale Husemöller, Michael Joachim, Branislav Jurčo, Martin Schottenloher, Section 15 of: Basic Bundle Theory and K-Cohomology Invariants, Springer Lecture Notes in Physics 726, 2008, (pdf, doi:10.1007/978-3-540-74956-1)

   diff, v13, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeNov 23rd 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Eldon Dyer]], [[Richard Lashof]], *A topological Proof of the Bott Periodicity Theorems*, Annali di Matematica Pura ed Applicata **54** (1961) 231–254 &lbrack;[doi:10.1007/BF02415354](https://doi.org/10.1007/BF02415354)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Bott+periodicity/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Bott+periodicity/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Bott+periodicity">current</a>

   added pointer to:

   • Eldon Dyer, Richard Lashof, A topological Proof of the Bott Periodicity Theorems, Annali di Matematica Pura ed Applicata 54 (1961) 231–254 [doi:10.1007/BF02415354]

   diff, v15, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeNov 23rd 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexand pointer to the original: * [[Raoul Bott]], *The Stable Homotopy of the Classical Groups*, Proceedings of the National Academy of Sciences of the United States of America **43** 10 (1957) 933-935 &lbrack;[jstor:89403](https://www.jstor.org/stable/89403)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Bott+periodicity/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/Bott+periodicity/15">v15</a>, <a href="https://ncatlab.org/nlab/show/Bott+periodicity">current</a>

   and pointer to the original:

   • Raoul Bott, The Stable Homotopy of the Classical Groups, Proceedings of the National Academy of Sciences of the United States of America 43 10 (1957) 933-935 [jstor:89403]

   diff, v15, current