Related papers: A geometric description of the Atiyah-Hirzebruch s…
We construct the Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors' corresponding earlier construction for differential cohomology…
We construct a spectral sequence for computing KR-theory, analogous to the spectral sequence relating motivic cohomology to algebraic K-theory.
We propose that symmetry protected topological (SPT) phases with crystalline symmetry are formulated by equivariant generalized homologies $h^G_n(X)$ over a real space manifold $X$ with $G$ a crystalline symmetry group. The…
We study the Atiyah-Hirzebruch spectral sequence (AHSS) for equivariant K-theory in the context of band theory. Various notions in the band theory such as irreducible representations at high-symmetric points, the compatibility relation,…
We introduce the notion of Atiyah class of a generalized holomorphic vector bundle, which captures the obstruction to the existence of generalized holomorphic connections on the bundle. As in the classical holomorphic case, this Atiyah…
We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah-Hirzebruch (AHSS) type, where we provide a…
Generalizing the ideas of $\mathbb{Z}_k$-manifolds from Sullivan and stratifolds from Kreck, we define $\mathbb{Z}_k$-stratifolds. We show that the bordism theory of $\mathbb{Z}_k$-stratifolds is sufficient to represent all homology classes…
It has been proposed that cobordism and K-theory groups, which can be mathematically related in certain cases, are physically associated to generalised higher-form symmetries. As a consequence, they should be broken or gauged in any…
To address the need for a unified framework that incorporates Lie algebroid connections on both vector and principal bundles, this paper investigates a generalized Atiyah algebroid structure and its short exact sequence. Building on this…
We show that the Atiyah-Hirzebruch K-theory of spaces admits a canonical generalization for stratified spaces. For this we study algebraic constructions on stratified vector bundles. In particular the tangent bundle of a stratified manifold…
We discuss the relations between the Atiyah-Hirzebruch spectral sequence and the Gysin map for a multiplicative cohomology theory, on spaces having the homotopy type of a finite CW-complex. In particular, let us fix such a multiplicative…
Let S be an essentially smooth scheme over a field of characteristic exponent c. Let MGL and HZ denote the algebraic cobordism spectrum and the motivic cohomology spectrum over S, respectively. We show that the canonical map MGL/(a1, a2,…
Recently Maldacena, Moore, and Seiberg (MMS) have proposed a physical interpretation of the Atiyah-Hirzebruch spectral sequence, which roughly computes the K-homology groups that classify D-branes. We note that in IIB string theory, this…
In this note, we use Curtis's algorithm and the Lambda algebra to compute the algebraic Atiyah-Hirzebruch spectral sequence of the suspension spectrum of $\mathbb{R}P^\infty$ with the aid of a computer, which gives us its Adams $E_2$-page…
We show that, for a finite spectrum $X$, Spanier-Whitehead duality induces an isomorphism between the cohomological and homological Atiyah-Hirzebruch spectral sequences. As an application, it follows that Poincar\'e duality for a Poincar\'e…
The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded center of its homology category: the characteristic homomorphism. We interpret it as an edge homomorphism in a…
We construct connections and characteristic forms for principal bundles over groupoids and stacks in the differentiable, holomorphic and algebraic category using Atiyah sequences associated to transversal tangential distributions.
We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our theory in the generality of objects of a…
We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild-Serre spectral sequence for Lie…
A careful account is given of generalized equivariant homology theories on the category of topological pairs acted on by a group. In particular, upon restriction to the category of equivariant simplicial complexes, the equivalence of…