University of Metz, France
and
University of Paderborn, Germany
Foliated ρ-invariants
Doctoral thesis by
Indrava Roy
Under the Supervision of
Prof. Dr. Moulay-Tahar Benameur, University of Metz
Prof. Dr. Joachim Hilgert, University of Paderborn
Dissertation Committee:
Mathai Varghese, Adelaide (Referee)
Paolo Piazza, Rome (Referee)
James Heitsch, Chicago
Jean-Louis Tu, Metz
Bachir Bekka, Rennes
Moulay-Tahar Benameur, Metz (Supervisor)
Joachim Hilgert, Paderborn (Supervisor)
2
3
Acknowledgements
It is with great pleasure that I take this opportunity to thank everyone who has given me the strength
and support to complete this thesis. First and foremost I would like to express my deep gratitude towards
my thesis advisors Prof. Moulay Benameur and Prof. Joachim Hilgert, without their inspiration and help,
both professionally and personally, it is impossible for me to imagine completing this task. I consider myself
extremely lucky to have found such inspiring teachers.
I also thank Prof. Paolo Piazza and Prof. Mathai Varghese for agreeing to be referees for the thesis, and
the jury members, Prof. James Heitsch, Prof. Jean-Louis Tu and Prof. Bekka Bachir. I drew great strength
and inspiration from discussions with Prof. Paolo Piazza, Prof. Jean-Louis Tu, Prof. James Heitsch, Prof.
Nigel Higson, Prof. John Roe, Prof. Alexander Gorokhovsky, Prof. Thomas Schick, Prof. Herv´e Oyono-
Oyono, Prof. S. Rajeev, Dr. Alexander Alldridge and Dr. Troels Johansen. I sincerely thank Prof. Tilmann
Wurzbacher and Prof. Angela Pasquale for helping me in various administrative and personal reasons. I would
also like to thank my teachers in IIT Kharagpur, especially Prof. Pratima Panigrahi, Prof. S. Banerjee, Prof.
A. R. Roy and Prof. S.S. Alam. I find it humbling to have had the chance to learn and gain inspiration from
Prof. K. B. Sinha, Prof. V. S. Sundar, and Prof. Peter Fillmore. It was Prof. Fillmore who inspired me to
take up research in Mathematics. I also cherish the many friends I made during conferences and workshops,
especially Paolo Antonini, Paolo Carillo, Alessandro Fermi, Pierre Clare, Haija Mostafa, Uuye Otogonbayar
and Maria Paula Gomez.
Among my colleagues from Paderborn, I can’t thank enough Nikolay Dichev, Carsten Balleier, Anke Pohl, Jan
M¨ollers and Stefan Wolf and Sameh Keliny for making me feel like home in the Department and for countless
other ways in which they helped me, both professionally and personally. Among my friends in Paderborn, I
must mention “Mahomies” Su, Martin, Cheng Yee, Erik, Dalimir, Melisa, Maria, Mariana, Fran, Timo and
Luz-Marina for making it feel like a home away from home and for so many wonderful memories to cherish.
No expression of thanks suffices for the wonderful people at PACE, especially Frau Canisius, who made it so
much easier to deal with administrative problems in her magically efficient ways. I would also like to express
my gratitude for Herr Das and Frau Hartmann who helped me in my neediest hour.
I take this opportunity to thank my colleagues from Metz, Mathieu, Sara, Mike, Fred, Elkaioum, Alexandre,
Ivan, Wael, among others, who helped me innumerable times in administrative, personal, and professional
matters. I especially thank Mathieu and Sara for helping me grasp many mathematical concepts with their
sharp insights during our many enlightening discussions. I cannot express enough my gratitude for Fred
and Elkaioum who helped me at crucial times when my poor French got in the way of things. Outside the
professional circles, I express my deepest thanks to Jawad, Saba, Sundar, Praveena, Ananthi and Sajid for
creating a homely environment in Metz and their generosity in lending a helping hand at the drop of a hat.
Last, but certainly not the least, it would be a futile attempt to express in words my gratitude to my family.
This day would not have come without the love and support of my wife, Saumya. She has been a source
of strength all throughout, especially when the chips were down. The influence of my parents and my elder
brother to achieve my dreams can never be fathomed. Their constant encouragement and support saw me
through many a difficult time. It is my pleasure to dedicate this thesis to all of them.
4
R´esum´e de la th`ese en fran¸cais
Dans cette th`ese, nous avons introduit, pour un feuilletage mesur´e donn´e, les ” rho-invariants feuillet´e ” pour
lesquels nous nous sommes attach´es de prouver certaines de leurs propri´et´es de stabilit´e. En particulier,
nous avons d´emontr´e que le rho-invariant associ´e `a l’op´erateur de signature est ind´ependant de la m´etrique
consid´er´ee sur le feuilletage, ainsi que son invariance par rapport aux “ diff´eomorphismes de feuilletage ”, ce
que g´en´eralise un r´esultat classique de Cheeger et Gromov.
Nous avons ´egalement obtenu une g´en´eralisation du th´eorme du Gamma-indice d’Atiyah pour les feuilletages.
Ce r´esultat est d´ej`a connu des experts, mais une preuve d´etaill´ee n’est pas disponible dans la litt´erature. De
plus, nous avons ´etendu le formalisme des complexes de Hilbert-Poincar´e (HP) aux cas des feuilletages, et
avons construit une ´equivalence d’homotopie explicite pour les HP-complexes sur des feuilletages ´equivalents
par homotopie feuillet´e. Cela nous permet en particulier de donner une preuve directe d’un r´esultat d´ej`a
connu sur l’invariance par homotopie de la classe “ signature d’indice ” pour les feuilletages. Enfin, nous
indiquons, comme application de ce formalisme, comment prolonger partiellement la preuve de l’invariance
par homotopie sur les rho-invariants classiques de Cheeger et Gromov.
Zusammenfassung der Dissertation auf Deutsch
Wir f¨uhren in dieser Dissertation die foliated rho-Invarianten auf measured Bl¨atterungen ein und beweisen
einige Stabilittseigenschaften. Wir beweisen insbesondere, dass die “foliated rho-Invariante” metrisch un-
abh¨angig und invariant unter Diffeomorphismen ist. Dies ist eine Erweiterung eines klassischen Resultats
von Cheeger und Gromov. Wir erreichen so eine Verallgemeinerung des Gamma-Index Theorems von Atiyah
f¨ur Foliations, die Experten bekannt, aber nicht in der Literatur zu finden war. Wir erweitern den Hilbert-
Poincar (HP) Komplex Formalismus fr den Fall von Bl¨atterungen und konstruieren eine explizite Homo-
topie¨aquivalenz von HP-Komplexen auf leafwise Homotopie ¨aquivalenten Bl¨atterungen. Das liefert einen
direkten Beweis des bereits bekannten Resultats ¨uber die Homotopieinvarianz der Signaturindexklasse f¨ur
Bl¨atterungen. Wir geben zuletzt eine Anwendung dieses Formalismus, um den Beweis der Homotopieinvari-
anz der klassischen Cheeger-Gromov rho-Invarianten teilweise auf den foliated Fall zu erweitern.
Contents
1 Introduction 9
1.1 Part I: Foliated Atiyah’s theorem and the Baum-Connes map . . . . . . . . . . . . . . . . . . 11
1.2 Part II: Stability properties of the foliated ρ-invariant . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Part III: Hilbert-Poincar´e complexes for foliations . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Part IV: Application to homotopy invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Background on foliations and Operator algebras 17
2.1 Foliated Charts and Foliated Atlases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.2 Groupoids associated to a foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Noncommutative integration theory on foliations . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Tangential measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Transverse measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Integrating a tangential measure against a holonomy invariant measure . . . . . . . . 22
2.3 Operator algebras on foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 The convolution algebra on a groupoid . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Representations of Bc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.3 C∗-algebra of a foliation with coefficients in a vector bundle . . . . . . . . . . . . . . . 24
2.3.4 Von Neumann Algebras for foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.5 Traces on foliations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Foliated Atiyah’s theorem 31
3.1 Pseudodifferential operators on Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1 Longitudinal Pseudodifferential operators on Foliations and its monodromy groupoid . 31
3.1.2 Almost local pseudodifferential operators on foliations . . . . . . . . . . . . . . . . . . 34
5
Loading more pages...