New York Journal of Mathematics
Volume 28 (2022), 1256-1294


Sara Azzali, Sarah L. Browne, Maria Paula Gomez Aparicio, Lauren C. Ruth, and Hang Wang

K-homology and K-theory of pure braid groups

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Published: September 8, 2022.
Keywords: pure braid, K-theory, K-homology, Baum-Connes conjecture.
Subject [2010]: 58B34, 19D55, 46L80, 20F36.

We produce an explicit description of the K-theory and K-homology of the pure braid group on n strands. We describe the Baum--Connes correspondence between the generators of the left- and right-hand sides for n=4. Using functoriality of the assembly map and direct computations, we recover Oyono-Oyono's result on the Baum--Connes conjecture for pure braid groups [24]. We also discuss the case of the full braid group on 3-strands.


We thank Alain Valette for the suggestion to examine K-theory and K-homology of pure braid groups. We thank the organizers of the Women in Operator Algebras Conference that took place at BIRS where this project started. HW acknowledges the support from Science and Technology Commission of Shanghai Municipality (STCSM), grant No.18dz2271000. MGA was partially supported by ANR project Singstar.

Author information

Sara Azzali:
Universitá degli Studi di Bari
Dipartimento di Matematica
Via E. Orabona 4, 70125 Bari, Italy


Sarah L. Browne:
The University of Kansas
Department of Mathematics
1460 Jayhawk Blvd, Lawrence, KS 66045, USA


Maria Paula Gomez Aparicio:
Université Paris-Saclay, CNRS
Laboratoire de mathématiques d'Orsay
91405, Orsay, France


Lauren C. Ruth:
Mercy College
555 Broadway
Dobbs Ferry, NY 10522, USA


Hang Wang:
School of Mathematical Sciences and Shanghai Key Laboratory of PMMP
East China Normal University
Shanghai 200241, China