no. 6
Equivariant Callias index theory via coarse geometry
[Théorie d’indice Callias équivariant par géométrie grossière]
Guo, Hao1 ; Hochs, Peter2 ; Mathai, Varghese3
1 Texas A&M University Department of Mathematics Mailstop 3368 College Station TX 77843–3368 (United States)
2 Radboud University Department of Mathematics Institute for Mathematics, Astrophysics and Particle Physics Heyendaalseweg 135 6525 AJ Nijmegen (The Netherlands)
3 The University of Adelaide School of Mathematical Sciences SA 5005 (Australia)
Annales de l'Institut Fourier, Tome 71 (2021) no. 6, pp. 2387-2430.

L’indice grossier équivariant est bien compris et utilisé pour les actions par les groupes discrets. On commence par étendre la définition de cet indice aux groupes localement compacts généraux. On utilise une notion de modules admissibles sur des C * -algèbres de functions continues, pour obtenir un indice utile. Inspirés par le travail de Roe, nous développons une variante localisée, à valeurs dans la K-théorie de la C * -algèbre d’un groupe, généralisant l’assembly map de Baum–Connes aux actions non-cocompactes. On montre qu’un indice pour des opérateurs de type Callias est un cas spécial de cet indice localisé ; on obtient des résultats sur l’existence et la non-existence de métriques Riemanniennes à courbure scalaire positive, invariantes par des actions propres ; et on montre qu’une version localisée de la conjecture de Baum–Connes est plus faible que la conjecture originale, et on donne une description conceptuelle de la K-théorie des C * -algèbres de groupes.

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We first extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over C * -algebras of continuous functions to obtain a meaningful index. Inspired by a work of Roe, we then develop a localised variant, with values in the K-theory of a group C * -algebra. This generalises the Baum–Connes assembly map to non-cocompact actions. We show that an equivariant index for Callias-type operators is a special case of this localised index, obtain results on existence and non-existence of Riemannian metrics of positive scalar curvature invariant under proper group actions, and show that a localised version of the Baum–Connes conjecture is weaker than the original conjecture, while still giving a conceptual description of the K-theory of a group C * -algebra.

Reçu le :
Révisé le :
Accepté le :
Première publication :
Publié le :
DOI : 10.5802/aif.3445
Classification : 19K56, 58J20, 46L80, 22D15
Keywords: Roe algebra, equivariant index, proper group action, locally compact group, Callias-type operator
Mot clés : Algèbre de Roe, indice équivariant, action propre, groupe localement compact, opérateur de type Callias
Guo, Hao 1 ; Hochs, Peter 2 ; Mathai, Varghese 3

1 Texas A&M University Department of Mathematics Mailstop 3368 College Station TX 77843–3368 (United States)
2 Radboud University Department of Mathematics Institute for Mathematics, Astrophysics and Particle Physics Heyendaalseweg 135 6525 AJ Nijmegen (The Netherlands)
3 The University of Adelaide School of Mathematical Sciences SA 5005 (Australia)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2021__71_6_2387_0,
     author = {Guo, Hao and Hochs, Peter and Mathai, Varghese},
     title = {Equivariant {Callias} index theory via coarse geometry},
     journal = {Annales de l'Institut Fourier},
     pages = {2387--2430},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {71},
     number = {6},
     year = {2021},
     doi = {10.5802/aif.3445},
     zbl = {07554450},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3445/}
}
TY  - JOUR
AU  - Guo, Hao
AU  - Hochs, Peter
AU  - Mathai, Varghese
TI  - Equivariant Callias index theory via coarse geometry
JO  - Annales de l'Institut Fourier
PY  - 2021
SP  - 2387
EP  - 2430
VL  - 71
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3445/
DO  - 10.5802/aif.3445
LA  - en
ID  - AIF_2021__71_6_2387_0
ER  - 
%0 Journal Article
%A Guo, Hao
%A Hochs, Peter
%A Mathai, Varghese
%T Equivariant Callias index theory via coarse geometry
%J Annales de l'Institut Fourier
%D 2021
%P 2387-2430
%V 71
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3445/
%R 10.5802/aif.3445
%G en
%F AIF_2021__71_6_2387_0
Guo, Hao; Hochs, Peter; Mathai, Varghese. Equivariant Callias index theory via coarse geometry. Annales de l'Institut Fourier, Tome 71 (2021) no. 6, pp. 2387-2430. doi : 10.5802/aif.3445. https://aif.centre-mersenne.org/articles/10.5802/aif.3445/

[1] Abels, Herbert Parallelizability of proper actions, global K-slices and maximal compact subgroups, Math. Ann., Volume 212 (1974), pp. 1-19 | DOI | MR | Zbl

[2] Anghel, Nicolae Remark on Callias’ index theorem, Rep. Math. Phys., Volume 28 (1989) no. 1, pp. 1-6 | DOI | MR | Zbl

[3] Anghel, Nicolae On the index of Callias-type operators, Geom. Funct. Anal., Volume 3 (1993) no. 5, pp. 431-438 | DOI | MR | Zbl

[4] Baum, Paul; Connes, Alain; Higson, Nigel Classifying space for proper actions and K-theory of group C * -algebras, C * -algebras: 1943–1993 (San Antonio, 1993) (Contemporary Mathematics), Volume 167, American Mathematical Society, 1994, pp. 240-291 | DOI | MR | Zbl

[5] Bott, Raoul; Seeley, Robert Some remarks on the paper of Callias, Commun. Math. Phys., Volume 62 (1978) no. 3, pp. 235-245 | DOI | MR | Zbl

[6] Braverman, Maxim The index theory on non-compact manifolds with proper group action, J. Geom. Phys., Volume 98 (2015), pp. 275-284 | DOI | MR | Zbl

[7] Braverman, Maxim; Cecchini, Simone Callias-type operators in von Neumann algebras, J. Geom. Anal., Volume 28 (2018) no. 1, pp. 546-586 | DOI | MR | Zbl

[8] Brüning, Jochen; Moscovici, Henri L 2 -index for certain Dirac-Schrödinger operators, Duke Math. J., Volume 66 (1992) no. 2, pp. 311-336 | DOI | MR | Zbl

[9] Bunke, Ulrich A K-theoretic relative index theorem and Callias-type Dirac operators, Math. Ann., Volume 303 (1995) no. 2, pp. 241-279 | DOI | MR | Zbl

[10] Bunke, Ulrich; Engel, Alexander The coarse index class with support (2018) (https://arxiv.org/abs/1706.06959)

[11] Callias, Constantine Axial anomalies and index theorems on open spaces, Commun. Math. Phys., Volume 62 (1978) no. 3, pp. 213-234 | DOI | MR | Zbl

[12] Carvalho, Catarina; Nistor, Victor An index formula for perturbed Dirac operators on Lie manifolds, J. Geom. Anal., Volume 24 (2014) no. 4, pp. 1808-1843 | DOI | MR | Zbl

[13] Cecchini, Simone Callias-type operators in C * -algebras and positive scalar curvature on noncompact manifolds, J. Topol. Anal., Volume 12 (2020) no. 4, pp. 897-939 | DOI | MR | Zbl

[14] Connes, Alain; Moscovici, Henri The L 2 -index theorem for homogeneous spaces of Lie groups, Ann. Math., Volume 115 (1982) no. 2, pp. 291-330 | DOI | MR | Zbl

[15] Gong, Guihua; Wang, Qin; Yu, Guoliang Geometrization of the strong Novikov conjecture for residually finite groups, J. Reine Angew. Math., Volume 621 (2008), pp. 159-189 | DOI | MR | Zbl

[16] Gromov, Mikhael; Lawson, H. Blaine Jr. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math., Inst. Hautes Étud. Sci. (1984) no. 58, pp. 83-196 | MR | Zbl

[17] Guo, Hao Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature, J. Geom. Anal., Volume 31 (2021) no. 1, pp. 1-34 | DOI | MR | Zbl

[18] Guo, Hao; Hochs, Peter; Mathai, Varghese Coarse geometry and Callias quantisation, Trans. Am. Math. Soc., Volume 374 (2021) no. 4, pp. 2479-2520 | DOI | MR | Zbl

[19] Guo, Hao; Mathai, Varghese; Wang, Hang Positive scalar curvature and Poincaré duality for proper actions, J. Noncommut. Geom., Volume 13 (2019) no. 4, pp. 1381-1433 | DOI | MR | Zbl

[20] Guo, Hao; Xie, Zhizhang; Yu, Guoliang A Lichnerowicz Vanishing Theorem for the Maximal Roe Algebra (2021) (https://arxiv.org/abs/1905.12299)

[21] Haagerup, Uffe; Przybyszewska, Agata Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces (2006) (https://arxiv.org/abs/math/0606794)

[22] Higson, Nigel; Roe, John Analytic K-homology, Oxford Mathematical Monographs, Oxford University Press, 2000, xviii+405 pages | MR

[23] Higson, Nigel; Roe, John; Yu, Guoliang A coarse Mayer–Vietoris principle, Math. Proc. Camb. Philos. Soc., Volume 114 (1993) no. 1, pp. 85-97 | DOI | MR | Zbl

[24] Hochs, Peter; Mathai, Varghese Geometric quantization and families of inner products, Adv. Math., Volume 282 (2015), pp. 362-426 | DOI | MR | Zbl

[25] Hochs, Peter; Song, Yanli An equivariant index for proper actions III: The invariant and discrete series indices, Differ. Geom. Appl., Volume 49 (2016), pp. 1-22 | DOI | MR | Zbl

[26] Hochs, Peter; Wang, Bai-Ling; Wang, Hang An equivariant Atiyah–Patodi–Singer index theorem for proper actions I: the index formula (2020) (https://arxiv.org/abs/1904.11146)

[27] Hochs, Peter; Wang, Bai-Ling; Wang, Hang An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index (2020) (https://arxiv.org/abs/2006.08086)

[28] Kottke, Chris An index theorem of Callias type for pseudodifferential operators, J. K-Theory, Volume 8 (2011) no. 3, pp. 387-417 | DOI | MR | Zbl

[29] Kottke, Chris A Callias-type index theorem with degenerate potentials, Commun. Partial Differ. Equations, Volume 40 (2015) no. 2, pp. 219-264 | DOI | MR | Zbl

[30] Kramer, W. The scalar curvature on totally geodesic fiberings, Ann. Global Anal. Geom., Volume 18 (2000) no. 6, pp. 589-600 | DOI | MR | Zbl

[31] Kucerovsky, Dan A short proof of an index theorem, Proc. Am. Math. Soc., Volume 129 (2001) no. 12, pp. 3729-3736 | DOI | MR | Zbl

[32] O’Neill, Barrett The fundamental equations of a submersion, Mich. Math. J., Volume 13 (1966), pp. 459-469 | MR | Zbl

[33] Roe, John Coarse cohomology and index theory on complete Riemannian manifolds, Mem. Am. Math. Soc., Volume 104 (1993) no. 497, p. x+90 | DOI | MR | Zbl

[34] Roe, John Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics, 90, American Mathematical Society, 1996, x+100 pages | DOI | MR

[35] Roe, John Comparing analytic assembly maps, Q. J. Math, Volume 53 (2002) no. 2, pp. 241-248 | DOI | MR | Zbl

[36] Roe, John Lectures on coarse geometry, University Lecture Series, 31, American Mathematical Society, 2003, viii+175 pages | DOI | MR

[37] Roe, John Positive curvature, partial vanishing theorems and coarse indices, Proc. Edinb. Math. Soc., II. Ser., Volume 59 (2016) no. 1, pp. 223-233 | DOI | MR | Zbl

[38] Schick, Thomas The topology of positive scalar curvature, Proceedings of the International Congress of Mathematicians (Seoul 2014) Vol. II (2014), pp. 1285-1307 | MR | Zbl

[39] Vilms, Jaak Totally geodesic maps, J. Differ. Geom., Volume 4 (1970), pp. 73-79 | MR | Zbl

[40] Willett, Rufus; Yu, Guoliang Higher index theory, Cambridge Studies in Advanced Mathematics, 189, Cambridge University Press, 2020, xi+581 pages | DOI

[41] Wimmer, Robert An index for confined monopoles, Commun. Math. Phys., Volume 327 (2014) no. 1, pp. 117-149 | DOI | MR | Zbl

[42] Yu, Guoliang The Novikov conjecture for groups with finite asymptotic dimension, Ann. Math., Volume 147 (1998) no. 2, pp. 325-355 | DOI | MR | Zbl

[43] Yu, Guoliang The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., Volume 139 (2000) no. 1, pp. 201-240 | DOI | MR | Zbl

[44] Yu, Guoliang A characterization of the image of the Baum–Connes map, Quanta of maths (Clay Mathematics Proceedings), Volume 11, American Mathematical Society, 2010, pp. 649-657 | MR | Zbl

Cité par Sources :