Mathematics

We define Hochschild and cyclic homologies for bornological coarse spaces: for a fixed field k and group G , these are lax symmetric monoidal functors XH H G k and XH C G k from the category of equivariant bornological coarse spaces GBornCoarse to the cocomplete stable ∞ -category of chain complexes Ch ∞ . We relate these equivariant coarse homology theories to coarse algebraic K -theory X K G k and to coarse ordinary homology X H G by constructing a trace-like natural transformation X K G k →X H G that factors through coarse Hochschild (or cyclic) homology. We further compare the forget-control map for coarse Hochschild homology with the associated generalized assembly map.

We study graded connected algebras over a field of characteristic zero and give an explicit formula for the cyclic homology of a tensor algebra. By means of a slightly new definition of David Anick's notion "strongly free" we are able to prove that cyclic homology of an algebra of global dimension two is zero in homological degree greater than one and is zero also in homological degree equal to one in case the relations are monomials. We give also explicit formulas for the cyclic homology of a tensor algebra modulo one symmetric quadratic form.

A theorem of Pfister asserts that every 12 -dimensional quadratic form with trivial discriminant and trivial Clifford invariant over a field of characteristic different from 2 decomposes as a tensor product of a binary quadratic form and a 6 -dimensional quadratic form with trivial discriminant. The main result of the paper extends Pfister's result to orthogonal involutions: every central simple algebra of degree 12 with orthogonal involution of trivial discriminant and trivial Clifford invariant decomposes into a tensor product of a quaternion algebra and a central simple algebra of degree 6 with orthogonal involutions. This decomposition is used to establish a criterion for the existence of orthogonal involutions with trivial invariants on algebras of degree 12 , and to calculate the f 3 -invariant of the involution if the algebra has index 2 .

In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the K 0 -group of its group C ∗ -algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the Baum-Connes conjecture holds for a group, then Lott's delocalized eta invariants take values in algebraic numbers. We also generalize Lott's delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group.

The first main result of this paper is to prove that the convergence of Lott's delocalized eta invariant holds for all differential operators with a sufficiently large spectral gap at zero. Furthermore, to each delocalized cyclic cocycle, we define a higher analogue of Lott's delocalized eta invariant and prove its convergence when the delocalized cyclic cocycle has at most exponential growth. Our second main result is to obtain an explicit formula of the delocalized Connes-Chern character of all C ∗ -algebraic secondary invariants for word hyperbolic groups. Equivalently, we give an explicit formula for the pairing between C ∗ -algebraic secondary invariants and delocalized cyclic cocycles of the group algebra. When the C ∗ -algebraic secondary invariant is a K -theoretic higher rho invariant of an invertible differential operator, we show this pairing is precisely the higher analogue of Lott's delocalized eta invariant alluded to above. Our work uses Puschnigg's smooth dense subalgebra for word hyperbolic groups in an essential way. We emphasize that our construction of the delocalized Connes-Chern character is at C ∗ -algebra K -theory level. This is of essential importance for applications to geometry and topology. As a consequence, we compute the paring between delocalized cyclic cocycles and C ∗ -algebraic Atiyah-Patodi-Singer index classes for manifolds with boundary, when the fundamental group of the given manifold is hyperbolic.

In this short note, for a morphism of Waldhausen categories f:A=(A, w A )→B=(B, w B ) , we will define Conef to be a Waldhausen category. There exists the canonical morphism of Waldhausen categories κ f :B→Conef . We will show that the sequence A → f B → κ f Conef induces fibration sequence of spaces K(A) → K(f) K(B) → K( κ f ) K(Conef) on connective K -theory. Moreover we will define a notion of strictly derivable Waldhausen categories and define non-connective K -theory for strictly derivable Waldhausen categories.

We define derived Poincaré--Birkhoff--Witt maps of dg operads or derived PBW maps, for short, which extend the definition of PBW maps between operads of V.~Dotsenko and the second author in 1804.06485, with the purpose of studying the universal enveloping algebra of dg Lie algebras as a functor on the homotopy category. Our main result shows that the map from the homotopy Lie operad to the homotopy associative operad is derived PBW, which gives us an amenable description of the homology of the universal envelope of an L ∞ -algebra in the sense of Lada--Markl. We deduce from this several known results involving universal envelopes of L ∞ -algebras of V. Baranovsky and J. Moreno-Fernández, and extend D. Quillen's classical quasi-isomorphism C⟶BU from dg Lie algebras to L ∞ -algebras; this confirms a conjecture of J. Moreno-Fernández.

We introduce a derived representation scheme associated with a quiver, which may be thought of as a derived version of a Nakajima variety. We exhibit an explicit model for the derived representation scheme as a Koszul complex and by doing so we show that it has vanishing higher homology if and only if the moment map defining the corresponding Nakajima variety is flat. In this case we prove a comparison theorem relating isotypical components of the representation scheme to equivariant K-theoretic classes of tautological bundles on the Nakajima variety. As a corollary of this result we obtain some integral formulas present in the mathematical and physical literature since a few years, such as the formula for Nekrasov partition function for the moduli space of framed instantons on S 4 . On the technical side we extend the theory of relative derived representation schemes by introducing derived partial character schemes associated with reductive subgroups of the general linear group and constructing an equivariant version of the derived representation functor for algebras with a rational action of an algebraic torus.

We prove that derived equivalent algebras have isomorphic differential calculi in the sense of Tamarkin--Tsygan.

We prove derived invariance of the cap product for associative algebras projective over a commutative ring.