2010, 17: 138-154. doi: 10.3934/era.2010.17.138

The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations

1. 

Institut für Mathematik, Humboldt Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

2. 

Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, United States

3. 

Department of Mathematics, Texas Christian University, Fort Worth, Texas 76129, United States

Received  August 2010 Revised  October 2010 Published  December 2010

In this expository paper, we explain a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications is an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.
Citation: Jochen Brüning, Franz W. Kamber, Ken Richardson. The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations. Electronic Research Announcements, 2010, 17: 138-154. doi: 10.3934/era.2010.17.138
References:
[1]

P. Albin and R. Melrose, Equivariant cohomology and resolution,, preprint \arXiv{0907.3211v2} [math.DG]., ().

[2]

M. F. Atiyah, "Elliptic Operators and Compact Groups,", Lecture Notes in Mathematics \textbf{401}, 401 (1974).

[3]

M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. I,, Math. Proc. Camb. Phil. Soc., 77 (1975), 43. doi: 10.1017/S0305004100049410.

[4]

M. F. Atiyah and G. B. Segal, The index of elliptic operators. II,, Ann. of Math. (2), 87 (1968), 531. doi: 10.2307/1970716.

[5]

N. Berline, E. Getzler and M. Vergne, "Heat Kernels and Dirac Operators,", Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 298 (1992).

[6]

N. Berline and M. Vergne, The Chern character of a transversally elliptic symbol and the equivariant index,, Invent. Math., 124 (1996), 11. doi: 10.1007/s002220050045.

[7]

N. Berline and M. Vergne, L'indice équivariant des opérateurs transversalement elliptiques,, (French) [The equivariant index of transversally elliptic operators], 124 (1996), 51. doi: 10.1007/s002220050046.

[8]

B. Booss-Bavnbek and K. P. Wojciechowski, "Elliptic Boundary Problems for Dirac Operators,", Mathematics: Theory & Applications. Birkhäuser Boston, (1993).

[9]

G. Bredon, "Introduction to Compact Transformation Groups,", Pure and Applied Mathematics, (1972).

[10]

J. Brüning and E. Heintze, Representations of compact Lie groups and elliptic operators,, Inv. Math., 50 (): 169. doi: 10.1007/BF01390288.

[11]

J. Brüning and E. Heintze, The asymptotic expansion of Minakshisundaram-Pleijel in the equivariant case,, Duke Math. Jour., 51 (1984), 959.

[12]

J. Brüning, F. W. Kamber and K. Richardson, The eta invariant and equivariant index of transversally elliptic operators,, preprint \arXiv{1005.3845v1} [math.DG]., ().

[13]

J. Brüning, F. W. Kamber and K. Richardson, Index theory for basic Dirac operators on Riemannian foliations,, preprint \arXiv{1008.1757v1} [math.DG]., ().

[14]

H. Donnelly, Eta invariants for $G$-spaces,, Indiana Univ. Math. J., 27 (1978), 889. doi: 10.1512/iumj.1978.27.27060.

[15]

A. El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications,, (French) [Transversely elliptic operators on a Riemannian foliation, 73 (1990), 57.

[16]

A. El Kacimi, G. Hector and V. Sergiescu, La cohomologie basique d'un feuilletage riemannien est de dimension finie,, (French) [The basic cohomology of a Riemannian foliation is finite-dimensional], 188 (1985), 593. doi: 10.1007/BF01161658.

[17]

J. F. Glazebrook and F. W. Kamber, Transversal Dirac families in Riemannian foliations,, Comm. Math. Phys., 140 (1991), 217. doi: 10.1007/BF02099498.

[18]

A. Gorokhovsky and J. Lott, The index of a transverse Dirac-type operator: The case of abelian Molino sheaf,, preprint \arXiv{1005.0161v2} [math.DG]., ().

[19]

G. Habib and K. Richardson, A brief note on the spectrum of the basic Dirac operator,, Bull. London Math. Soc., 41 (2009), 683. doi: 10.1112/blms/bdp042.

[20]

F. W. Kamber and K. Richardson, $G$-equivariant vector bundles on manifolds of one orbit type,, preprint., ().

[21]

F. W. Kamber and Ph. Tondeur, "Foliated Bundles and Characteristic Classes,", Lecture Notes in Math. \textbf{493}, 493 (1975).

[22]

F. W. Kamber and Ph. Tondeur, Foliations and metrics,, Differential geometry (College Park, 32 (1981), 103.

[23]

F. W. Kamber and Ph. Tondeur, Duality for Riemannian foliations,, Singularities, (1981), 609.

[24]

F. W. Kamber and Ph. Tondeur, De Rham-Hodge theory for Riemannian foliations,, Math. Ann., 277 (1987), 415. doi: 10.1007/BF01458323.

[25]

K. Kawakubo, "The Theory of Transformation Groups,", Translated from the 1987 Japanese edition. The Clarendon Press, (1987).

[26]

T. Kawasaki, The index of elliptic operators over $V$ -manifolds,, Nagoya Math. J., 84 (1981), 135.

[27]

H. B. Lawson and M.-L. Michelsohn, "Spin Geometry,", Princeton Mathematical Series, (1989).

[28]

P. Molino, "Riemannian Foliations,", Translated from the French by Grant Cairns. With appendices by Cairns, (1988).

[29]

I. Prokhorenkov and K. Richardson, Natural equivariant Dirac operators,, to appear in Geom. Dedicata, ().

[30]

B. Reinhart, "Differential Geometry of Foliations -- The Fundamental Integrability Problem,", Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], (1983).

[31]

Ph. Tondeur, "Geometry of Foliations,", Monographs in Mathematics, 90 (1997).

show all references

References:
[1]

P. Albin and R. Melrose, Equivariant cohomology and resolution,, preprint \arXiv{0907.3211v2} [math.DG]., ().

[2]

M. F. Atiyah, "Elliptic Operators and Compact Groups,", Lecture Notes in Mathematics \textbf{401}, 401 (1974).

[3]

M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry. I,, Math. Proc. Camb. Phil. Soc., 77 (1975), 43. doi: 10.1017/S0305004100049410.

[4]

M. F. Atiyah and G. B. Segal, The index of elliptic operators. II,, Ann. of Math. (2), 87 (1968), 531. doi: 10.2307/1970716.

[5]

N. Berline, E. Getzler and M. Vergne, "Heat Kernels and Dirac Operators,", Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 298 (1992).

[6]

N. Berline and M. Vergne, The Chern character of a transversally elliptic symbol and the equivariant index,, Invent. Math., 124 (1996), 11. doi: 10.1007/s002220050045.

[7]

N. Berline and M. Vergne, L'indice équivariant des opérateurs transversalement elliptiques,, (French) [The equivariant index of transversally elliptic operators], 124 (1996), 51. doi: 10.1007/s002220050046.

[8]

B. Booss-Bavnbek and K. P. Wojciechowski, "Elliptic Boundary Problems for Dirac Operators,", Mathematics: Theory & Applications. Birkhäuser Boston, (1993).

[9]

G. Bredon, "Introduction to Compact Transformation Groups,", Pure and Applied Mathematics, (1972).

[10]

J. Brüning and E. Heintze, Representations of compact Lie groups and elliptic operators,, Inv. Math., 50 (): 169. doi: 10.1007/BF01390288.

[11]

J. Brüning and E. Heintze, The asymptotic expansion of Minakshisundaram-Pleijel in the equivariant case,, Duke Math. Jour., 51 (1984), 959.

[12]

J. Brüning, F. W. Kamber and K. Richardson, The eta invariant and equivariant index of transversally elliptic operators,, preprint \arXiv{1005.3845v1} [math.DG]., ().

[13]

J. Brüning, F. W. Kamber and K. Richardson, Index theory for basic Dirac operators on Riemannian foliations,, preprint \arXiv{1008.1757v1} [math.DG]., ().

[14]

H. Donnelly, Eta invariants for $G$-spaces,, Indiana Univ. Math. J., 27 (1978), 889. doi: 10.1512/iumj.1978.27.27060.

[15]

A. El Kacimi-Alaoui, Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications,, (French) [Transversely elliptic operators on a Riemannian foliation, 73 (1990), 57.

[16]

A. El Kacimi, G. Hector and V. Sergiescu, La cohomologie basique d'un feuilletage riemannien est de dimension finie,, (French) [The basic cohomology of a Riemannian foliation is finite-dimensional], 188 (1985), 593. doi: 10.1007/BF01161658.

[17]

J. F. Glazebrook and F. W. Kamber, Transversal Dirac families in Riemannian foliations,, Comm. Math. Phys., 140 (1991), 217. doi: 10.1007/BF02099498.

[18]

A. Gorokhovsky and J. Lott, The index of a transverse Dirac-type operator: The case of abelian Molino sheaf,, preprint \arXiv{1005.0161v2} [math.DG]., ().

[19]

G. Habib and K. Richardson, A brief note on the spectrum of the basic Dirac operator,, Bull. London Math. Soc., 41 (2009), 683. doi: 10.1112/blms/bdp042.

[20]

F. W. Kamber and K. Richardson, $G$-equivariant vector bundles on manifolds of one orbit type,, preprint., ().

[21]

F. W. Kamber and Ph. Tondeur, "Foliated Bundles and Characteristic Classes,", Lecture Notes in Math. \textbf{493}, 493 (1975).

[22]

F. W. Kamber and Ph. Tondeur, Foliations and metrics,, Differential geometry (College Park, 32 (1981), 103.

[23]

F. W. Kamber and Ph. Tondeur, Duality for Riemannian foliations,, Singularities, (1981), 609.

[24]

F. W. Kamber and Ph. Tondeur, De Rham-Hodge theory for Riemannian foliations,, Math. Ann., 277 (1987), 415. doi: 10.1007/BF01458323.

[25]

K. Kawakubo, "The Theory of Transformation Groups,", Translated from the 1987 Japanese edition. The Clarendon Press, (1987).

[26]

T. Kawasaki, The index of elliptic operators over $V$ -manifolds,, Nagoya Math. J., 84 (1981), 135.

[27]

H. B. Lawson and M.-L. Michelsohn, "Spin Geometry,", Princeton Mathematical Series, (1989).

[28]

P. Molino, "Riemannian Foliations,", Translated from the French by Grant Cairns. With appendices by Cairns, (1988).

[29]

I. Prokhorenkov and K. Richardson, Natural equivariant Dirac operators,, to appear in Geom. Dedicata, ().

[30]

B. Reinhart, "Differential Geometry of Foliations -- The Fundamental Integrability Problem,", Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], (1983).

[31]

Ph. Tondeur, "Geometry of Foliations,", Monographs in Mathematics, 90 (1997).

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