Raoul Bott (changes) in nLab

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Raoul Bott (1923–2005) was one of the great 20th century topologists and geometers. Among his famous works, one should mention thetopologists and geometers. Among his famous works, one should mention the Bott periodicity theorem (of importance in K-theory), studies in Morse theory (including the study of Bott–Morse functions), the Borel–Weil–Bott theorem in geometric representation theory, the study of fixed point (localization) formulas (the Atiyah–Bott fixed point theorem) and the Atiyah-Bott-Patodi slick proof of the index theorem via the heat kernel expansion.

• Wikipedia entry

• Loring Tu: The life and works of Raoul BottThe life and works of Raoul Bott (2002) [[harvard history](https://legacy-www.math.harvard.edu/history/bott/bottbio/index.html),celebratio:302]

• Michael Atiyah: Raoul Harry Bott: 24 September 1923 – 20 December 2005, Celebratio Mathematica (2007)

Selected writings

Introducing the concept and terminology of the Pontrjagin product on the ordinary homology of based loop spaces:

• Raoul Bott, Hans Samelson, On the Pontryagin product in spaces of paths, Commentarii Mathematici Helvetici 27 (1953) 320–337 [[doi:10.1007/BF02564566](https://doi.org/10.1007/BF02564566)]

and application to the case of Lie groups:

• Raoul Bott, The space of loops on a Lie group, Michigan Math. J. 5 1 (1958) 35-61 [[doi:10.1307/mmj/1028998010](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-5/issue-1/The-space-of-loops-on-a-Lie-group/10.1307/mmj/1028998010.full)]

Introducing Bott periodicity:

• 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](https://www.jstor.org/stable/89403)]

Introducing the Atiyah-Bott-Shapiro orientation MSpin→\toKO and MSpin^<i>c</i>→\toKU:

• Michael Atiyah, Raoul Bott, Arnold Shapiro, Clifford modules, Topology Volume 3, Supplement 1, July 1964, Pages 3-38 (doi:10.1016/0040-9383(64)90003-5, pdf)

Introducing the Atiyah-Bott fixed point theorem:

• Michael F. Atiyah, Raoul Bott: A Lefschetz fixed point formula for elliptic differential operators, Bull. Amer. Math. Soc. 72 (1966) 245-250 [[doi:10.1090/S0002-9904-1966-11483-0](https://doi.org/10.1090/S0002-9904-1966-11483-0), pdf]

• Michael F. Atiyah, Raoul Bott: A Lefschetz Fixed Point Formula for Elliptic Complexes: I, Annals of Mathematics 86 2 (1967) 374-407 [[doi:10.2307/1970694](https://doi.org/10.2307/1970694), jstor:1970694]

On topological K-theory:

• Raoul Bott, Lectures on K(X)K(X), Benjamin (1969) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/bottk.pdf), pdf]

  Russian transl. by B. Yu. Sternin, Matematika 11 2 (1967) 32–56 [[mathnet:mat424](https://www.mathnet.ru/eng/mat424)]

On the Chern-Weil homomorphism:

• Raoul Bott, On the Chern-Weil homomorphism and the continuous cohomology of Lie-groups, Advances in Mathematics Volume 11, Issue 3, December 1973, Pages 289-303 (doi:10.1016/0001-8708(73)90012-1)

On differential forms in algebraic topology:

• Raoul Bott, Loring Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer 1982. xiv+331 pp. (doi:10.1007/978-1-4757-3951-0, pdf)

On the simplicial de Rham complex and equivariant de Rham cohomology:

• Raoul Bott, Herbert Shulman, Jim Stasheff, On the de Rham theory of certain classifying spaces, Advances in Mathematics, Volume 20, Issue 1, April 1976, Pages 43-56 (doi:10.1016/0001-8708(76)90169-9, pdf)

On the rigidity theorem for elliptic genera:

• Raoul Bott, Clifford Taubes, On the Rigidity Theorems of Witten, Journal of the American Mathematical Society Vol. 2, No. 1 (Jan., 1989), pp. 137-186 (doi:10.2307/1990915)

• Raoul Bott, On the Fixed Point Formula and the Rigidity Theorems of Witten, Lectures at Cargése 1987. In: ’t Hooft G., Jaffe A., Mack G., Mitter P.K., Stora R. (eds) Nonperturbative Quantum Field Theory. NATO ASI Series (Series B: Physics), vol 185. Springer (1988) (doi:10.1007/978-1-4613-0729-7_2)