Mathematics

We adapt Grayson's model of higher algebraic K -theory using binary acyclic complexes to the setting of stable ∞ -categories. As an application, we prove that the K -theory of stable ∞ -categories preserves infinite products.

This article is the first of two where we investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras L(E) and L(F) of graphs E and F over a commutative ground ring ℓ . In this first article we consider Leavitt path algebras of general graphs over general ground rings; the second article will focus mostly on purely infinite simple unital Leavitt path algebras over a field. Bivariant algebraic K -theory kk is the universal homology theory with the properties above; we prove a structure theorem for unital Leavitt path algebras in kk . We show that under very mild assumptions on ℓ , for a graph E with finitely many vertices and reduced incidence matrix A E , the structure of L(E) depends only on the isomorphism classes of the cokernels of the matrix I− A E and of its transpose, which are respectively the kk groups K H 1 (L(E))=k k −1 (L(E),ℓ) and K H 0 (L(E))=k k 0 (ℓ,L(E)) . Hence if L(E) and L(F) are unital Leavitt path algebras such that K H 0 (L(E))≅K H 0 (L(F)) and K H 1 (L(E))≅K H 1 (L(F)) then no homology theory with the above properties can distinguish them. We also prove that for Leavitt path algebras, kk has several properties similar to those that Kasparov's bivariant K -theory has for C ∗ -graph algebras, including analogues of the Universal coefficient and Künneth theorems of Rosenberg and Schochet.

Building on Kadeishvili's original theorem inducing A ∞ -algebra structures on the homology of dg-algebras, several directions of algorithmic research in A ∞ -algebras have been pursued. In this paper we will survey work done on calculating explicit A ∞ -algebra structures from homotopy retractions; in group cohomology; and in persistent homology.

In this paper we introduce the notion of almost flatness for (stably) relative bundles on a pair of topological spaces and investigate basic properties of it. First, we show that almost flatness of topological and smooth sense are equivalent. This provides a construction of an almost flat stably relative bundle by using the enlargeability of manifolds. Second, we show the almost monodromy correspondence, that is, a correspondence between almost flat (stably) relative bundles and (stably) relative quasi-representations of the fundamental group.

In their construction of the topological index for flat vector bundles, Atiyah, Patodi and Singer associate to each flat vector bundle a particular C/Z - K -theory class. This assignment determines a map, up to weak homotopy, from K a C , the algebraic K -theory space of the complex numbers, to F t,C/Z , the homotopy fiber of the Chern character. In this paper, we give evidence for the conjecture that this map can be represented by an infinite loop map. The result of the paper implies a refined Bismut-Lott index theorem for a compact smooth bundle E→B with the fundamental group π 1 (E,∗) finite for every point ∗∈E .

Let R be a ring with unit. Passing to the colimit with respect to the standard inclusions GL(n,R)→GL(n+1,R) (which add a unit vector as new last row and column) yields, by definition, the stable linear group GL(R) ; the same result is obtained, up to isomorphism, when using the "opposite" inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic K -group K 1 (R)=GL(R)/E(R) of~ R , giving an elementary description that does not involve elementary matrices explicitly.

Consider a proper, isometric action by a unimodular locally compact group G on a Riemannian manifold M with boundary, such that M/G is compact. Then an equivariant Dirac-type operator D on M under a suitable boundary condition has an equivariant index index G (D) in the K -theory of the reduced group C ∗ -algebra C ∗ r G of G . This is a common generalisation of the Baum-Connes analytic assembly map and the (equivariant) Atiyah-Patodi-Singer index. In part I of this series, a numerical index index g (D) was defined for an element g∈G , in terms of a parametrix of D and a trace associated to g . An Atiyah-Patodi-Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions, τ g ( index G (D))= index g (D) , for a trace τ g defined by the orbital integral over the conjugacy class of g . This implies that the index theorem from part I yields information about the K -theoretic index index G (D) . It also shows that index g (D) is a homotopy-invariant quantity.

We prove an equivariant localized and norm-controlled version of the Pimsner-Popa-Voiculescu theorem. As an application, we deduce a proof of the Paschke-Higson duality for transformation groupoids.

Let G be a linear Lie group acting properly and isometrically on a G -spin c manifold M with compact quotient. We show that Poincaré duality holds between G -equivariant K -theory of M , defined using finite-dimensional G -vector bundles, and G -equivariant K -homology of M , defined through the geometric model of Baum and Douglas.

For a proper, cocompact action by a locally compact group of the form H×G , with H compact, we define an H×G -equivariant index of H -transversally elliptic operators, which takes values in K K ∗ ( C ∗ H, C ∗ G) . This simultaneously generalises the Baum--Connes analytic assembly map, Atiyah's index of transversally elliptic operators, and Kawasaki's orbifold index. This index also generalises the assembly map to elliptic operators on orbifolds. In the special case where the manifold in question is a real semisimple Lie group, G is a cocompact lattice and H is a maximal compact subgroup, we realise the Dirac induction map from the Connes--Kasparov conjecture as a Kasparov product and obtain an index theorem for Spin-Dirac operators on compact locally symmetric spaces.