M. F. Atiyah and G. B. Segal :
“The index of elliptic operators, II ,”
Uspehi Mat. Nauk
23 : 6 (144)
(1968 ),
pp. 135–149 .
Russian translation of article in Ann. Math. 87 :3 (1968) .
MR
0236953
article

People
BibTeX
@article {key0236953m,
AUTHOR = {Atiyah, M. F. and Segal, G. B.},
TITLE = {The index of elliptic operators, {II}},
JOURNAL = {Uspehi Mat. Nauk},
FJOURNAL = {Uspekhi Matematicheskikh Nauk},
VOLUME = {23},
NUMBER = {6 (144)},
YEAR = {1968},
PAGES = {135--149},
NOTE = {Russian translation of article in \textit{Ann.
Math.} \textbf{87}:3 (1968). MR:0236953.},
ISSN = {0042-1316},
}
M. F. Atiyah and G. B. Segal :
“The index of elliptic operators, II ,”
Ann. Math. (2)
87 : 3
(1968 ),
pp. 531–545 .
Russian translation published in Uspehi Mat. Nauk 23 :6(144) (1968) .
MR
0236951
Zbl
0164.24201
article

Abstract
People
BibTeX

The purpose of this paper is to show how the index theorem of [Atiyah and Singer 1963] can be reformulated as a general “Lefschetz fixed-point theorem” on the lines of [Atiyah and Bott 1967]. In this way we shall obtain the main theorem of [Atiyah and Bott 1967], generalized to deal with arbitrary fixed-point sets, but only for transformations belonging to a compact group.

The content of this paper is essentially topological, and it should be viewed as a paper on the equivariant \( K \) -theory of manifolds. The analysis has all been done in [Atiyah and Singer 1968], and what we do here is simply to express the topological index in terms of fixed-point sets. This is quite independent of the main theorem of [Atiyah and Singer 1968] asserting the equality of the topological and analytical indices.

As in [Atiyah and Singer 1968], we avoid cohomology and use only \( K \) -theory. In paper III of this series, we shall pass over to cohomology obtaining explicit formulas in terms of characteristic classes.

@article {key0236951m,
AUTHOR = {Atiyah, M. F. and Segal, G. B.},
TITLE = {The index of elliptic operators, {II}},
JOURNAL = {Ann. Math. (2)},
FJOURNAL = {Annals of Mathematics, second series},
VOLUME = {87},
NUMBER = {3},
YEAR = {1968},
PAGES = {531--545},
DOI = {10.2307/1970716},
NOTE = {Russian translation published in \textit{Uspehi
Mat. Nauk} \textbf{23}:6(144) (1968).
MR:0236951. Zbl:0164.24201.},
ISSN = {0003-486X},
}
M. F. Atiyah and G. B. Segal :
“Equivariant \( K \) -theory and completion ,”
J. Differential Geometry
3
(1969 ),
pp. 1–18 .
MR
0259946
Zbl
0215.24403
article

Abstract
People
BibTeX

It was shown in [Atiyah 1961] that, for any finite group \( G \) , the completed character ring \( R(G)^{\wedge} \) was isomorphic to \( K^*(B_G) \) where \( B_G \) denotes a classifying space for \( G \) . The corresponding result for compact connected Lie groups was established in [Atiyah and Hirzebruch 1961], and a combination of the methods of [Atiyah and Hirzebruch 1961] and [Atiyah 1961] (together with certain basic properties of \( R(G) \) given in [Segal 1968b]) can be used to derive the theorem for general compact Lie groups. Such a proof however would be extremely lengthy, the worst part being in fact the treatment for finite groups where one climbs up via cyclic and Sylow subgroups.

The purpose of this paper is to give a new and much simpler proof of the theorem about \( K^*(B_G) \) which applies directly to all compact Lie groups \( G \) . The main feature of our new proof is that we generalize the whole problem in a rather natural way by working with the equivariant \( K \) -theory developed in [Segal 1968a]. We shall formulate and prove a general theorem about the completion \( K_G^*(X)^{\wedge} \) for any compact \( G \) -space \( X \) . The theorem about \( R(G) \) then follows by taking \( X \) to be a point.

@article {key0259946m,
AUTHOR = {Atiyah, M. F. and Segal, G. B.},
TITLE = {Equivariant \$K\$-theory and completion},
JOURNAL = {J. Differential Geometry},
FJOURNAL = {Journal of Differential Geometry},
VOLUME = {3},
YEAR = {1969},
PAGES = {1--18},
URL = {http://projecteuclid.org/euclid.jdg/1214428815},
NOTE = {MR:0259946. Zbl:0215.24403.},
ISSN = {0022-040X},
}
M. F. Atiyah and G. B. Segal :
“Exponential isomorphisms for \( \lambda \) -rings ,”
Quart. J. Math. Oxford Ser. (2)
22 : 3
(1971 ),
pp. 371–378 .
MR
0291250
Zbl
0226.13008
article

Abstract
People
BibTeX
@article {key0291250m,
AUTHOR = {Atiyah, M. F. and Segal, G. B.},
TITLE = {Exponential isomorphisms for \$\lambda\$-rings},
JOURNAL = {Quart. J. Math. Oxford Ser. (2)},
FJOURNAL = {The Quarterly Journal of Mathematics,
second series},
VOLUME = {22},
NUMBER = {3},
YEAR = {1971},
PAGES = {371--378},
DOI = {10.1093/qmath/22.3.371},
NOTE = {MR:0291250. Zbl:0226.13008.},
ISSN = {0033-5606},
}
M. Atiyah and G. Segal :
“On equivariant Euler characteristics ,”
J. Geom. Phys.
6 : 4
(1989 ),
pp. 671–677 .
MR
1076708
Zbl
0708.19004
article

Abstract
People
BibTeX
@article {key1076708m,
AUTHOR = {Atiyah, Michael and Segal, Graeme},
TITLE = {On equivariant {E}uler characteristics},
JOURNAL = {J. Geom. Phys.},
FJOURNAL = {Journal of Geometry and Physics},
VOLUME = {6},
NUMBER = {4},
YEAR = {1989},
PAGES = {671--677},
DOI = {10.1016/0393-0440(89)90032-6},
NOTE = {MR:1076708. Zbl:0708.19004.},
ISSN = {0393-0440},
}
M. Atiyah and G. Segal :
“Twisted \( K \) -theory ,”
Ukr. Mat. Visn.
1 : 3
(2004 ),
pp. 287–330 .
MR
2172633
Zbl
1151.55301
article

People
BibTeX
@article {key2172633m,
AUTHOR = {Atiyah, Michael and Segal, Graeme},
TITLE = {Twisted \$K\$-theory},
JOURNAL = {Ukr. Mat. Visn.},
FJOURNAL = {Ukra\"ins'kyj Matematychnyj Visnyk},
VOLUME = {1},
NUMBER = {3},
YEAR = {2004},
PAGES = {287--330},
NOTE = {MR:2172633. Zbl:1151.55301.},
ISSN = {1810-3200},
}
M. Atiyah and G. Segal :
“Twisted \( K \) -theory and cohomology ,”
pp. 5–43
in
Inspired by S. S. Chern .
Edited by P. Griffiths .
Nankai Tracts in Mathematics 11 .
World Scientific (Hackensack, NJ ),
2006 .
MR
2307274
Zbl
1138.19003
ArXiv
math/0510674
incollection

Abstract
People
BibTeX

We explore the relations of twisted \( K \) -theory to twisted and untwisted classical cohomology. We construct an Atiyah–Hirzebruch spectral sequence, and describe its differentials rationally as Massey products. We define the twisted Chern character. We also discuss power operations in the twisted theory, and the role of the Koschorke classes.

@incollection {key2307274m,
AUTHOR = {Atiyah, Michael and Segal, Graeme},
TITLE = {Twisted \$K\$-theory and cohomology},
BOOKTITLE = {Inspired by {S}.~{S}. {C}hern},
EDITOR = {Phillip Griffiths},
SERIES = {Nankai Tracts in Mathematics},
NUMBER = {11},
PUBLISHER = {World Scientific},
ADDRESS = {Hackensack, NJ},
YEAR = {2006},
PAGES = {5--43},
NOTE = {ArXiv:math/0510674. MR:2307274. Zbl:1138.19003.},
ISBN = {9789812700629},
}
G. Segal :
“Being a graduate student of Michael Atiyah ,”
pp. 47–48
in
The founders of index theory: Reminiscences of and about Sir Michael Atiyah, Raoul Bott, Friedrich Hirzebruch, and I. M. Singer ,
2nd edition.
Edited by S.-T. Yau .
International Press (Somerville, MA ),
2009 .
Also in the 2003 edition of the book.
incollection

People
BibTeX
@incollection {key94049467,
AUTHOR = {Segal, Graeme},
TITLE = {Being a graduate student of {M}ichael
{A}tiyah},
BOOKTITLE = {The founders of index theory: {R}eminiscences
of and about {S}ir {M}ichael {A}tiyah,
{R}aoul {B}ott, {F}riedrich {H}irzebruch,
and {I}.~{M}. {S}inger},
EDITOR = {Yau, Shing-Tung},
EDITION = {2nd},
PUBLISHER = {International Press},
ADDRESS = {Somerville, MA},
YEAR = {2009},
PAGES = {47--48},
NOTE = {Also in the 2003 edition of the book.},
ISBN = {9781571461377},
}