Homology, Homotopy and Applications

Hyperoctahedral homology for involutive algebras is the homology theory associated to the hyperoctahedral crossed simplicial group. It is related to equivariant stable homotopy theory via the homology of equivariant infinite loop spaces. In this paper we show that there is an E-infinity algebra structure on the simplicial module that computes hyperoctahedral homology. We deduce that hyperoctahedral homology admits Dyer–Lashof homology operations. Furthermore, there is a Pontryagin product which gives hyperoctahedral homology the structure of an associative, graded-commutative algebra. We also give an explicit description of hyperoctahedral homology in degree zero. Combining this description and the Pontryagin product we show that hyperoctahedral homology fails to preserve Morita equivalence.

Let $\pi : X \to S$ be a family of smooth projective curves, and let $L$ and $M$ be a pair of line bundles on $X$. We show that Deligne’s line bundle $\langle L,M \rangle$ can be obtained from the $\mathcal{K}_2$-gerbe $G_{L,M}$ constructed in [AR16] via an integration along the fiber map for gerbes that categorifies the well known one arising from the Leray spectral sequence of $\pi$. Our construction provides a full account of the biadditivity properties of $\langle L,M \rangle$. Our main application is to the categorification of correspondences on the self-product of a curve.

We show that there is an equivalence in any $n$-topos $\mathscr{X}$ between the pointed and $k$-connective objects of $\mathscr{X}$ and the $\mathbb{E}_k$-group objects of the $(n-k-1)$-truncation of $\mathscr{X}$. This recovers, up to equivalence of $\infty$-categories, some classical results regarding algebraic models for $k$-connective, $(n-1)$-coconnective homotopy types. Further, it extends those results to the case of sheaves of such homotopy types. We also show that for any pointed and $k$-connective object $X$ of $\mathscr{X}$ there is an equivalence between the $\infty$-category of modules in $\mathscr{X}$ over the associative algebra $\Omega^k X$, and the $\infty$-category of comodules in $\mathscr{X}$ for the cocommutative coalgebra $\Omega^{k-1} X$. All of these equivalences are given by truncations of Lurie’s $\infty$-categorical bar and cobar constructions, hence the terminology “Koszul duality.”

We prove that every locally Cartesian closed $\infty$-category with a subobject classifier has a strict initial object and disjoint and universal binary coproducts.

In this paper, we calculate the coefficient ring of equivariant Thom complex cobordism for the symmetric group on three elements. We also make some remarks on general methods of calculating certain pullbacks of rings which typically occur in calculations of equivariant cobordism.

We classify invertible $2$-dimensional framed and $r$-spin topological field theories by computing the homotopy groups and the $k$-invariant of the corresponding bordism categories. The zeroth homotopy group of a bordism category is the usual Thom bordism group, the first homotopy group can be identified with a Reinhart vector field bordism group, or the so called SKK group as observed by Ebert, Bökstedt–Svane and Kreck–Stolz–Teichner. We present the computation of SKK groups for stable tangential structures. Then we consider non-stable examples: the $2$-dimensional framed and $r$-spin SKK groups and compute them explicitly using the combinatorial model of framed and $r$-spin surfaces of Novak, Runkel and the author.

The rational homology of unordered configuration spaces of points on any surface was studied by Drummond–Cole and Knudsen. We compute the rational cohomology of configuration spaces on a closed orientable surface, keeping track of the mixed Hodge numbers and the action of the symplectic group on the cohomology. We find a series with coefficients in the Grothendieck ring of $\mathfrak{sp}(2g)$ that describes explicitly the decomposition of the cohomology into irreducible representations. From that we deduce the mixed Hodge numbers and the Betti numbers, obtaining a new formula without cancellations.

There is a free construction from multicategories to permutative categories, left adjoint to the endomorphism multicategory construction. The main result shows that these functors induce an equivalence of homotopy theories. This result extends a similar result of Thomason, that permutative categories model all connective spectra.

We construct a comparison map from the topological $K$-theory of the dg-category of twisted perfect complexes on certain global quotient stacks to twisted equivariant $K$-theory, generalizing constructions of Halpern-Leistner–Pomerleano [HLP15] and Moulinos [Mou19]. We prove that this map is an equivalence if a version of the projective bundle theorem holds for twisted equivariant $K$-theory. Along the way, we give a new proof of a theorem of Moulinos that the comparison map is an equivalence in the non-equivariant case.

We define an obstruction to the formality of a differential graded algebra over a graded operad defined over a commutative ground ring. This obstruction lives in the derived operadic cohomology of the algebra. Moreover, it determines all operadic Massey products induced on the homology algebra, hence the name of derived universal Massey product.