M. F. Atiyah and F. Hirzebruch :
“Riemann–Roch theorems for differentiable manifolds ,”
Bull. Amer. Math. Soc.
65 : 4
(1959 ),
pp. 276–281 .
MR
0110106
Zbl
0142.40901
article
Abstract
People
BibTeX
The Riemann–Roch Theorem for an algebraic variety \( Y \) (see [Hirzebruch 1956]) led to certain divisibility conditions for the Chern classes of \( Y \) . It was natural to ask whether these conditions held more generally for any compact almost complex manifold. This question, and various generalizations of it, were raised in [Hirzebruch 1954] and most of these have since been answered in the affirmative in [Borel and Hirzebruch 1958; Milnor 1960].
More recently Grothendieck [Borel and Serre 1958] has obtained a more general Riemann–Roch Theorem for a map \( R: Y \rightarrow X \) of algebraic varieties. This reduces to the previous Riemann–Roch Theorem on taking \( X \) to be a point. Grothendieck’s Theorem implies many conditions on characteristic classes, and again it is natural to ask if these conditions hold more generally for almost complex or even differentiable manifolds. The purpose of this note is to enunciate certain differentiable analogues of Grothendieck’s Theorem. These “differentiable Riemann–Roch Theorems” yield, as special cases, the divisibility conditions mentioned above and also certain new homotopy invariance properties of Pontrjagin classes. As an application of the latter we get a new proof (and slight improvement) of the result of Kervaire–Milnor [1958] on the stable \( J \) -homomorphism.
@article {key0110106m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Riemann--{R}och theorems for differentiable
manifolds},
JOURNAL = {Bull. Amer. Math. Soc.},
VOLUME = {65},
NUMBER = {4},
YEAR = {1959},
PAGES = {276--281},
DOI = {10.1090/S0002-9904-1959-10344-X},
NOTE = {MR:0110106. Zbl:0142.40901.},
ISSN = {0002-9904},
}
M. F. Atiyah and F. Hirzebruch :
“Quelques théorèmes de non-plongement pour les variétés différentiables ”
[Some theorems of non-immersion for differentiable manifolds ],
Bull. Soc. Math. France
87
(1959 ),
pp. 383–396 .
See also Colloques Int. Centre Nat. Rech. Sci. 89 (1960) .
MR
0114231
article
Abstract
People
BibTeX
Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].
La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958], [Borel and Hirzebruch 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espace euclidien (ou ce qui est équivalent, une sphère) de dimension \( 2d - q \) . Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui étaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott [Borel and Hirzebruch 1958/59; Bott 1958] qui dit que la \( n \) -ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \) .
@article {key0114231m,
AUTHOR = {Atiyah, Michael F. and Hirzebruch, Friedrich},
TITLE = {Quelques th\'eor\`emes de non-plongement
pour les vari\'et\'es diff\'erentiables
[Some theorems of non-immersion for
differentiable manifolds]},
JOURNAL = {Bull. Soc. Math. France},
FJOURNAL = {Bulletin de la Societe Mathematique
de France},
VOLUME = {87},
YEAR = {1959},
PAGES = {383--396},
URL = {http://www.numdam.org/item?id=BSMF_1959__87__383_0},
NOTE = {See also \textit{Colloques Int. Centre
Nat. Rech. Sci.} \textbf{89} (1960).
MR:0114231.},
ISSN = {0037-9484},
}
M. F. Atiyah and F. Hirzebruch :
“Quelques théoremes de non-plongement pour les variétés différentiables ”
[Some non-embedding theorems for differentiable manifolds ],
Colloques Int. Centre Nat. Rech. Sci.
89
(1960 ),
pp. 383–396 .
See also Bull. Soc. Math. France 87 (1959) .
Zbl
0108.18202
article
Abstract
People
BibTeX
Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch, 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].
La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958; 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espance euclidien (ou ce qui est équivalent, une spère) de dimension \( 2d - q \) . Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui ’\etaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott ([Borel and Hirzebruch, 1958/59] et [Bott 1958]) qui dit que la \( n \) -ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \) .
@article {key0108.18202z,
AUTHOR = {Atiyah, Michael F. and Hirzebruch, Friedrich},
TITLE = {Quelques th\'eoremes de non-plongement
pour les vari\'et\'es diff\'erentiables
[Some non-embedding theorems for differentiable
manifolds]},
JOURNAL = {Colloques Int. Centre Nat. Rech. Sci.},
FJOURNAL = {Colloques Internationaux du Centre National
de la Recherche Scientifique},
VOLUME = {89},
YEAR = {1960},
PAGES = {383--396},
NOTE = {See also \textit{Bull. Soc. Math. France}
\textbf{87} (1959). Zbl:0108.18202.},
ISSN = {0366-7634},
}
M. F. Atiyah and F. Hirzebruch :
“Vector bundles and homogeneous spaces ,”
pp. 7–38
in
Differential geometry .
Edited by C. B. Allendoerfer .
Proceedings of Symposia in Pure Mathematics 3 .
American Mathematical Society (Providence, RI ),
1961 .
MR
0139181
Zbl
0108.17705
incollection
People
BibTeX
@incollection {key0139181m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Vector bundles and homogeneous spaces},
BOOKTITLE = {Differential geometry},
EDITOR = {Carl Barnett Allendoerfer},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {3},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1961},
PAGES = {7--38},
NOTE = {MR:0139181. Zbl:0108.17705.},
ISSN = {0082-0717},
}
M. F. Atiyah and F. Hirzebruch :
“Cohomologie-Operationen und charakteristische Klassen ,”
Math. Z.
77 : 1
(1961 ),
pp. 149–187 .
MR
0156361
Zbl
0109.16002
article
People
BibTeX
@article {key0156361m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Cohomologie-{O}perationen und charakteristische
{K}lassen},
JOURNAL = {Math. Z.},
FJOURNAL = {Mathematische Zeitschrift},
VOLUME = {77},
NUMBER = {1},
YEAR = {1961},
PAGES = {149--187},
DOI = {10.1007/BF01180171},
NOTE = {MR:0156361. Zbl:0109.16002.},
ISSN = {0025-5874},
}
M. F. Atiyah and F. Hirzebruch :
“Bott periodicity and the parallelizability of the spheres ,”
Proc. Cambridge Philos. Soc.
57
(1961 ),
pp. 223–226 .
MR
0126282
Zbl
0108.35902
article
People
BibTeX
@article {key0126282m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Bott periodicity and the parallelizability
of the spheres},
JOURNAL = {Proc. Cambridge Philos. Soc.},
FJOURNAL = {Mathematical Proceedings of the Cambridge
Philosophical Society},
VOLUME = {57},
YEAR = {1961},
PAGES = {223--226},
NOTE = {MR:0126282. Zbl:0108.35902.},
ISSN = {0305-0041},
}
M. F. Atiyah and F. Hirzebruch :
“Charakteristische Klassen und Anwendungen ,”
Enseignement Math. (2)
7
(1961 ),
pp. 188–213 .
MR
0154294
Zbl
0104.39801
article
People
BibTeX
@article {key0154294m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Charakteristische {K}lassen und {A}nwendungen},
JOURNAL = {Enseignement Math. (2)},
FJOURNAL = {Enseignement Mathematique, second series},
VOLUME = {7},
YEAR = {1961},
PAGES = {188--213},
DOI = {10.5169/seals-37131},
NOTE = {MR:0154294. Zbl:0104.39801.},
ISSN = {0013-8584},
}
M. F. Atiyah and F. Hirzebruch :
“The Riemann–Roch theorem for analytic embeddings ,”
Topology
1 : 2
(1962 ),
pp. 151–166 .
MR
0148084
Zbl
0108.36402
article
Abstract
People
BibTeX
In [Borel and Serre 1958] Grothendieck formulated and proved a generalization of the Riemann–Roch theorem which we shall refer to as GRR. This theorem is concerned with a proper morphism \( f:Y \rightarrow X \) of algebraic manifolds (any ground field) and reduces to the version (HRR) given in [Hirzebruch 1956] when \( X \) is a point (and the ground field is \( \mathbb{C} \) ). It is not known whether GRR or even HRR holds for arbitrary complex manifolds. However, the proof of GRR given in [Borel and Serre 1958] breaks up into two separate cases:
\( f \) is an embedding,
\( f \) is a projection of \( X \times P_N \rightarrow X \) , where \( P_N \) is a projective space,
and the main purpose of this paper is to give a proof of GRR in case (i) for arbitrary complex manifolds. This proof is quite different from, and in many ways simpler than, that of [Borel and Serre 1958] and, for the complex algebraic case, it gives a new proof of GRR.
@article {key0148084m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {The {R}iemann--{R}och theorem for analytic
embeddings},
JOURNAL = {Topology},
FJOURNAL = {Topology},
VOLUME = {1},
NUMBER = {2},
YEAR = {1962},
PAGES = {151--166},
DOI = {10.1016/0040-9383(65)90023-6},
NOTE = {MR:0148084. Zbl:0108.36402.},
ISSN = {0040-9383},
}
M. F. Atiyah and F. Hirzebruch :
“Analytic cycles on complex manifolds ,”
Topology
1
(1962 ),
pp. 25–45 .
MR
0145560
Zbl
0108.36401
article
Abstract
People
BibTeX
Let \( X \) be a complex manifold, \( Y \) a closed irreducible \( k \) -dimensional complex analytic subspace of \( X \) . Then \( Y \) defines or “carries” a \( 2k \) -dimensional integral homology class \( y \) of \( X \) , although the precise definition of \( y \) presents technical difficulties. A finite formal linear combination \( \sum n_i Y_i \) with \( n_i \) integers and \( Y_i \) as above is called a complex analytic cycle, and the corresponding homology class \( \sum n_i y_i \) is called a complex analytic homology class. If an integral cohomology class \( u \) corresponds under Poincaré duality to a complex analytic homology class we shall say that \( u \) is a complex analytic cohomology class . The purpose of this paper is to show that a complex analytic cohomology class \( u \) satisfies certain topological conditions, independent of the complex structure of \( X \) . These conditions are that certain cohomology operations should vanish on \( u \) , for example \( \mathrm{Sq}^3u = 0 \) : they are all torsion conditions. We also produce examples to show that these conditions are not vacuous even in the restricted classes of (a) Stein manifolds and (b) projective algebraic manifolds.
@article {key0145560m,
AUTHOR = {Atiyah, M. F. and Hirzebruch, F.},
TITLE = {Analytic cycles on complex manifolds},
JOURNAL = {Topology},
FJOURNAL = {Topology},
VOLUME = {1},
YEAR = {1962},
PAGES = {25--45},
DOI = {10.1016/0040-9383(62)90094-0},
NOTE = {MR:0145560. Zbl:0108.36401.},
ISSN = {0040-9383},
}
M. Atiyah and F. Hirzebruch :
“Spin-manifolds and group actions ,”
pp. 18–28
in
Essays on topology and related topics: Mémoires dédiés à Georges de Rham .
Edited by A. Haefliger and R. Narasimhan .
Springer (New York ),
1970 .
MR
0278334
Zbl
0193.52401
incollection
People
BibTeX
@incollection {key0278334m,
AUTHOR = {Atiyah, Michael and Hirzebruch, Friedrich},
TITLE = {Spin-manifolds and group actions},
BOOKTITLE = {Essays on topology and related topics:
{M}\'emoires d\'edi\'es \`a {G}eorges
de {R}ham},
EDITOR = {Andr\'e Haefliger and Raghavan Narasimhan},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1970},
PAGES = {18--28},
NOTE = {MR:0278334. Zbl:0193.52401.},
}
M. Atiyah :
“Eigenvalues of the Dirac operator ,”
pp. 251–260
in
Arbeitstagung Bonn 1984
(Max-Planck-Institut für Mathematik, Bonn, 15–22 June 1984 ).
Edited by F. Hirzebruch, J. Schwermer, and S. Suter .
Lecture Notes in Mathematics 1111 .
Springer (Berlin ),
1985 .
MR
797424
Zbl
0568.53022
incollection
Abstract
People
BibTeX
In recent years mathematicians have learnt a great deal from physicists and in particular from the work of Edward Witten. In a recent preprint [1984], Vafa and Witten have proved some striking results about the eigenvalues of the Dirac operator, and this talk will present their results. I shall concentrate entirely on the mathematical parts of their preprint leaving aside the physical interpretation which is their main motivation.
@incollection {key797424m,
AUTHOR = {Atiyah, Michael},
TITLE = {Eigenvalues of the {D}irac operator},
BOOKTITLE = {Arbeitstagung {B}onn 1984},
EDITOR = {Friedrich Hirzebruch and Joachim Schwermer
and Silke Suter},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1111},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1985},
PAGES = {251--260},
DOI = {10.1007/BFb0084593},
NOTE = {(Max-Planck-Institut f\"ur Mathematik,
Bonn, 15--22 June 1984). MR:797424.
Zbl:0568.53022.},
ISSN = {0075-8434},
ISBN = {9780387151953},
}
M. Atiyah :
“Commentary on the article of Manin ,”
pp. 103–109
in
Arbeitstagung Bonn 1984
(Max-Planck-Institut für Mathematik, Bonn, 15–22 June 1984 ).
Edited by F. Hirzebruch, J. Schwermer, and S. Suter .
Lecture Notes in Mathematics 1111 .
Springer (Berlin ),
1985 .
The article is Yu. I. Manin, “New dimensions in geometry,” from the same volume.
MR
797417
Zbl
0595.53071
incollection
Abstract
People
BibTeX
Manin’s stimulating contribution to the 25th Arbeitstagung which, in his absence, I attempted to present, provided me with an opportunity of adding some further reflections of my own. This commentary, which is therefore a very personal response to Manin’s article, consists of very general and speculative remarks about large areas of contemporary mathematics. Such speculations are, for good reason, rarely put down on paper but the record of the 25th Arbeitstagung provides a rather singular occasion where ideas of this type may not be out of place.
@incollection {key797417m,
AUTHOR = {Atiyah, Michael},
TITLE = {Commentary on the article of {M}anin},
BOOKTITLE = {Arbeitstagung {B}onn 1984},
EDITOR = {Friedrich Hirzebruch and Joachim Schwermer
and Silke Suter},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1111},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1985},
PAGES = {103--109},
DOI = {10.1007/BFb0084586},
NOTE = {(Max-Planck-Institut f\"ur Mathematik,
Bonn, 15--22 June 1984). The article
is {Y}u.\ {I}. {M}anin, ``{N}ew dimensions
in geometry,'' from the same volume.
MR:797417. Zbl:0595.53071.},
ISSN = {0075-8434},
ISBN = {9780387151953},
}
M. Atiyah :
“The European Mathematical Society ,”
pp. 1–5
in
Miscellanea mathematica .
Edited by P. J. Hilton, F. Hirzebruch, and R. Remmert .
Springer (Berlin ),
1991 .
MR
1131114
incollection
People
BibTeX
@incollection {key1131114m,
AUTHOR = {Atiyah, Michael},
TITLE = {The {E}uropean {M}athematical {S}ociety},
BOOKTITLE = {Miscellanea mathematica},
EDITOR = {Peter John Hilton and Friedrich Hirzebruch
and Reinhold Remmert},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1991},
PAGES = {1--5},
NOTE = {MR:1131114.},
ISBN = {9783540541745},
}
M. Atiyah :
“Friedrich Hirzebruch: An appreciation ,”
pp. 1–5
in
Proceedings of the Hirzebruch 65 conference on algebraic geometry
(Bar-Ilan University, Ramat Gan, Israel, 2–7 May 1993 ).
Edited by M. Teicher .
Israel Mathematical Conference Proceedings 9 .
Bar-Ilan University (Ramat Gan, Israel ),
1996 .
MR
1360493
Zbl
0834.01012
incollection
People
BibTeX
@incollection {key1360493m,
AUTHOR = {Atiyah, Michael},
TITLE = {Friedrich {H}irzebruch: {A}n appreciation},
BOOKTITLE = {Proceedings of the {H}irzebruch 65 conference
on algebraic geometry},
EDITOR = {Teicher, Mina},
SERIES = {Israel Mathematical Conference Proceedings},
NUMBER = {9},
PUBLISHER = {Bar-Ilan University},
ADDRESS = {Ramat Gan, Israel},
YEAR = {1996},
PAGES = {1--5},
NOTE = {(Bar-Ilan University, Ramat Gan, Israel,
2--7 May 1993). MR:1360493. Zbl:0834.01012.},
ISSN = {0792-4119},
ISBN = {9789995594244},
}
M. Atiyah :
“Physics and geometry: A look at the last twenty years ,”
pp. 1–8
in
Algebraic geometry: Hirzebruch 70
(Warsaw, 11–16 May 1998 ).
Edited by P. Pragacz, M. Szurek, and J. Wiśniewski .
Contemporary Mathematics 241 .
American Mathematical Society (Providence, RI ),
1999 .
MR
1718133
Zbl
0945.14026
incollection
Abstract
People
BibTeX
These are notes from the special lecture given by Professor Michael Atiyah during the “Algebraic Geometry Conference: Hirzebruch 70.” The text concerns the interactions between Physics and Geometry in the last two decades, and the role of Professor F. Hirzebruch and his Bonn “Arbeitstagung” in these interactions.
@incollection {key1718133m,
AUTHOR = {Atiyah, Michael},
TITLE = {Physics and geometry: {A} look at the
last twenty years},
BOOKTITLE = {Algebraic geometry: {H}irzebruch 70},
EDITOR = {Pragacz, Piotr and Szurek, Micha{\l}
and Wi\'sniewski, Jaros\l aw},
SERIES = {Contemporary Mathematics},
NUMBER = {241},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1999},
PAGES = {1--8},
NOTE = {(Warsaw, 11--16 May 1998). MR:1718133.
Zbl:0945.14026.},
ISSN = {0271-4132},
ISBN = {9780821811498},
}
M. Atiyah :
“\( K \) -theory past and present ,”
pp. 411–417
in
Sitzungsberichte der Berliner Mathematischen Gesellschaft, 1997–2000 .
Berliner Mathematischen Gesellschaft ,
2001 .
MR
2091892
Zbl
1061.19500
ArXiv
math/0012213
incollection
Abstract
People
BibTeX
@incollection {key2091892m,
AUTHOR = {Atiyah, Michael},
TITLE = {\$K\$-theory past and present},
BOOKTITLE = {Sitzungsberichte der {B}erliner {M}athematischen
{G}esellschaft, 1997--2000},
PUBLISHER = {Berliner Mathematischen Gesellschaft},
YEAR = {2001},
PAGES = {411--417},
NOTE = {ArXiv:math/0012213. MR:2091892. Zbl:1061.19500.},
}
M. F. Atiyah :
“A personal history ,”
pp. 5–15
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 .
Republished in Atiyah’s Collected works , vol. 6 .
incollection
People
BibTeX
@incollection {key72270854,
AUTHOR = {Atiyah, M. F.},
TITLE = {A personal history},
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 = {5--15},
NOTE = {Republished in Atiyah's \textit{Collected
works}, vol.~6.},
ISBN = {9781571461377},
}