A. Borel and F. Hirzebruch :
“Characteristic classes and homogeneous spaces, I ,”
Am. J. Math.
80 : 2
(April 1958 ),
pp. 458–538 .
MR
102800
Zbl
0097.36401
article
Abstract
People
BibTeX
It is known that the characteristic classes of a real or complex vector bundle may be interpreted as elementary symmetric functions in certain variables, which are 1, 2 or 4 dimensional cohomology classes. If we consider the tangent bundle of the coset space \( G/U \) of a compact connected Lie group modulo a closed subgroup, it turns out that these variables may be identified with certain roots of \( G \) (or their squares). Our first purpose is to establish this connection between roots and characteristic classes, which is the basis of this paper, and to compute the characteristic classes of certain well-known homogeneous spaces. These results are then applied in particular to \( G/T \) (\( T \) maximal torus of \( G \) ), and to other algebraic homogeneous spaces, where they lead to relations between characteristic classes, Betti numbers, the Riemann–Roch theorem and representation theory; they are also used to discuss multiplicative properties of the Todd genus and other genera in fibre bunldes with \( G/U \) as fibre. As an application, we get a divisibility property of the Chern class of a complex vector bundle over an even dimensional sphere which yields some information about certain homotopy groups of Lie groups.
@article {key102800m,
AUTHOR = {Borel, A. and Hirzebruch, F.},
TITLE = {Characteristic classes and homogeneous
spaces, {I}},
JOURNAL = {Am. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {80},
NUMBER = {2},
MONTH = {April},
YEAR = {1958},
PAGES = {458--538},
DOI = {10.2307/2372795},
NOTE = {MR:102800. Zbl:0097.36401.},
ISSN = {0002-9327},
}
A. Borel and F. Hirzebruch :
“Characteristic classes and homogeneous spaces, II ,”
Am. J. Math.
81 : 2
(April 1959 ),
pp. 315–382 .
MR
110105
article
People
BibTeX
@article {key110105m,
AUTHOR = {Borel, A. and Hirzebruch, F.},
TITLE = {Characteristic classes and homogeneous
spaces, {II}},
JOURNAL = {Am. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {81},
NUMBER = {2},
MONTH = {April},
YEAR = {1959},
PAGES = {315--382},
DOI = {10.2307/2372747},
NOTE = {MR:110105.},
ISSN = {0002-9327},
}
A. Borel and F. Hirzebruch :
“Characteristic classes and homogeneous spaces, III ,”
Am. J. Math.
82 : 3
(July 1960 ),
pp. 491–504 .
MR
120664
article
People
BibTeX
@article {key120664m,
AUTHOR = {Borel, A. and Hirzebruch, F.},
TITLE = {Characteristic classes and homogeneous
spaces, {III}},
JOURNAL = {Am. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {82},
NUMBER = {3},
MONTH = {July},
YEAR = {1960},
PAGES = {491--504},
DOI = {10.2307/2372969},
NOTE = {MR:120664.},
ISSN = {0002-9327},
}
F. Hirzebruch :
Topological methods in algebraic geometry ,
3rd, enlarged edition.
Grundlehren der Mathematischen Wissenschaften 131 .
Springer (New York ),
1966 .
With an appendix by R. L. E. Schwarzenberger and an additional section by A. Borel.
English translation of 2nd German edition (1962) .
MR
202713
Zbl
0138.42001
book
People
BibTeX
@book {key202713m,
AUTHOR = {Hirzebruch, F.},
TITLE = {Topological methods in algebraic geometry},
EDITION = {3rd, enlarged},
SERIES = {Grundlehren der Mathematischen Wissenschaften},
NUMBER = {131},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1966},
PAGES = {x+232},
NOTE = {With an appendix by R. L. E. Schwarzenberger
and an additional section by A. Borel.
English translation of 2nd German edition
(1962). MR:202713. Zbl:0138.42001.},
ISSN = {0072-7830},
}
F. Hirzebruch :
Topologiceskie metody v algebraiceskoi geometrii
[Topological methods in algebraic geometry ].
Mir (Moscow ),
1973 .
with appendices by R. L. E. Schwarzenberger and A. Borel.
Russian translation of 2nd German edition (1962) .
Zbl
0266.14011
book
People
BibTeX
@book {key0266.14011z,
AUTHOR = {Hirzebruch, Friedrich},
TITLE = {Topologiceskie metody v algebraiceskoi
geometrii [Topological methods in algebraic
geometry]},
PUBLISHER = {Mir},
ADDRESS = {Moscow},
YEAR = {1973},
PAGES = {280},
NOTE = {with appendices by R. L. E. Schwarzenberger
and A. Borel. Russian translation of
2nd German edition (1962). Zbl:0266.14011.},
}
F. Hirzebruch :
Topological methods in algebraic geometry ,
3rd (2nd corrected printing) edition.
Grundlehren der Mathematischen Wissenschaften 131 .
Springer (Berlin ),
1978 .
with appendices by R. L. E. Schwarzenberger and A. Borel.
Republication of 1966 English translation of 2nd German edition (1962) . Reprinted in 1995 .
Zbl
0376.14001
book
People
BibTeX
@book {key0376.14001z,
AUTHOR = {Hirzebruch, Friedrich},
TITLE = {Topological methods in algebraic geometry},
EDITION = {3rd (2nd corrected printing)},
SERIES = {Grundlehren der Mathematischen Wissenschaften},
NUMBER = {131},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1978},
PAGES = {xi+234},
NOTE = {with appendices by R. L. E. Schwarzenberger
and A. Borel. Republication of 1966
English translation of 2nd German edition
(1962). Reprinted in 1995. Zbl:0376.14001.},
ISSN = {0072-7830},
ISBN = {9783540035251},
}
F. Hirzebruch :
Topological methods in algebraic geometry ,
3rd edition.
Classics in Mathematics 131 .
Springer (Berlin ),
1995 .
With appendices by R. L. E. Schwarzenberger and A. Borel, and a preface by the author and Schwarzenberger.
Reprint of the 1978 edition .
MR
1335917
Zbl
0843.14009
book
People
BibTeX
@book {key1335917m,
AUTHOR = {Hirzebruch, Friedrich},
TITLE = {Topological methods in algebraic geometry},
EDITION = {3rd},
SERIES = {Classics in Mathematics},
NUMBER = {131},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1995},
PAGES = {xii+234},
DOI = {10.1007/978-3-642-62018-8},
NOTE = {With appendices by R. L. E. Schwarzenberger
and A. Borel, and a preface by the author
and Schwarzenberger. Reprint of the
1978 edition. MR:1335917. Zbl:0843.14009.},
ISSN = {1431-0821},
ISBN = {9783540586630},
}
A. W. Knapp, J.-P. Serre, K. Chandrasekharan, E. Bombieri, F. Hirzebruch, T. A. Springer, J. Tits, J. Arthur, G. Prasad, and M. Goresky :
“Armand Borel ,”
Asian J. Math.
8 : 4
(2004 ),
pp. xlvii–xcvi .
Reprinted from Notices Amer. Math. Soc. 51 :5 (2004),
498–524.
MR
2127939
People
BibTeX
@article {key2127939m,
AUTHOR = {Knapp, Anthony W. and Serre, Jean-Pierre
and Chandrasekharan, Komaravolu and
Bombieri, Enrico and Hirzebruch, Friedrich
and Springer, Tonny A. and Tits, Jacques
and Arthur, James and Prasad, Gopal
and Goresky, Mark},
TITLE = {Armand {B}orel},
JOURNAL = {Asian J. Math.},
FJOURNAL = {The Asian Journal of Mathematics},
VOLUME = {8},
NUMBER = {4},
YEAR = {2004},
PAGES = {xlvii--xcvi},
NOTE = {Reprinted from \textit{Notices Amer.
Math. Soc.} \textbf{51}:5 (2004), 498--524.
MR 2127939.},
ISSN = {1093-6106},
}
J. Arthur, E. Bombieri, K. Chandrasekharan, F. Hirzebruch, G. Prasad, J.-P. Serre, T. A. Springer, and J. Tits :
“Armand Borel (1923–2003) ,”
Notices Am. Math. Soc.
51 : 5
(May 2004 ),
pp. 498–524 .
The various contributions were reprinted in Asian J. Math. 8 :4 (2004) .
MR
2046057
Zbl
1052.01012
article
People
BibTeX
@article {key2046057m,
AUTHOR = {Arthur, James and Bombieri, Enrico and
Chandrasekharan, Komaravolu and Hirzebruch,
Friedrich and Prasad, Gopal and Serre,
Jean-Pierre and Springer, Tonny A. and
Tits, Jacques},
TITLE = {Armand {B}orel (1923--2003)},
JOURNAL = {Notices Am. Math. Soc.},
FJOURNAL = {Notices of the American Mathematical
Society},
VOLUME = {51},
NUMBER = {5},
MONTH = {May},
YEAR = {2004},
PAGES = {498--524},
NOTE = {The various contributions were reprinted
in \textit{Asian J. Math.} \textbf{8}:4
(2004). MR:2046057. Zbl:1052.01012.},
ISSN = {0002-9920},
CODEN = {AMNOAN},
}
F. Hirzebruch :
“Armand Borel ,”
Asian J. Math.
8 : 4
(December 2004 ),
pp. lxiii–lxviii .
Reprinted from Notices Am. Math. Soc. 51 :5 (2004) .
Zbl
1072.01535
article
People
BibTeX
@article {key1072.01535z,
AUTHOR = {Hirzebruch, Friedrich},
TITLE = {Armand {B}orel},
JOURNAL = {Asian J. Math.},
FJOURNAL = {Asian Journal of Mathematics},
VOLUME = {8},
NUMBER = {4},
MONTH = {December},
YEAR = {2004},
PAGES = {lxiii--lxviii},
DOI = {10.4310/AJM.2004.v8.n4.a11},
NOTE = {Reprinted from \textit{Notices Am. Math.
Soc.} \textbf{51}:5 (2004). Zbl:1072.01535.},
ISSN = {1093-6106},
}
F. Hirzebruch :
“The projective tangent bundles of a complex three-fold .”
Edited by L. Ji, K. Liu, and S.-T. Yau .
Pure Appl. Math. Q.
1 : 3
(2005 ),
pp. 441–448 .
Dedicated to my friend Armand Borel in memory of the golden fifties in Princeton.
MR
2201324
Zbl
1187.14009
article
People
BibTeX
@article {key2201324m,
AUTHOR = {Hirzebruch, Friedrich},
TITLE = {The projective tangent bundles of a
complex three-fold},
JOURNAL = {Pure Appl. Math. Q.},
FJOURNAL = {Pure and Applied Mathematics Quarterly},
VOLUME = {1},
NUMBER = {3},
YEAR = {2005},
PAGES = {441--448},
DOI = {10.4310/PAMQ.2005.v1.n3.a2},
NOTE = {\textit{In memory of {A}rmand {B}orel,
part 2 of 3}. Issue edited by L. Ji,
K. Liu, and S.-T. Yau. Dedicated
to my friend Armand Borel in memory
of the golden fifties in Princeton.
MR:2201324. Zbl:1187.14009.},
ISSN = {1558-8599},
}