The bicategory of parametrized spectra has a remarkably rich structure. We can take traces in this bicategory, giving classical invariants that count fixed points. We can also take C_n -equivariant external smash powers and equivariant traces, which give significant generalizations of the classical invariants that count periodic points. Unfortunately, the existence of these smash powers and traces for parametrized spectra depends on technical statements about the bicategory that can be difficult to verify directly, especially if one wants the construction to have a direct geometric interpretation. In this paper, we demonstrate the effectiveness of two tools—rigidity and deformable functors—by using them to establish formal structures on this bicategory directly from point-set level data.
The Morava E -theories, $$E_{n}$$ E n , are complex-oriented 2-periodic ring spectra, with homotopy groups $$W_{{{\mathbb {F}}}_{p^{n}}}[[u_{1}, u_{2},\ldots , u_{n-1}]][u,u^{-1}]$$ W F p n [ [ u 1 , u 2 , … , u n - 1 ] ] [ u , u - 1 ] . Here W denotes the ring of Witt vectors. $$E_{n}$$ E n is a Landweber exact spectrum and hence uniquely determined by its homotopy groups as $$BP_{*}$$ B P ∗ -algebra. Algebraic K -theory of $$E_{n}$$ E n is a key ingredient towards analyzing the layers in the p -complete Waldhausen’s algebraic K -theory chromatic tower. One hopes to use the machinery of trace methods to get results towards algebraic K -theory once the computation for $$THH(E_{n})$$ T H H ( E n ) is known. In this paper we describe $$THH(E_{2})$$ T H H ( E 2 ) as part of consecutive chain of cofiber sequences where each cofiber sits in the next cofiber sequence and the first term of each cofiber sequence is describable completely in terms of suspensions and localizations of $$E_{2}$$ E 2 . For these results, we first calculate K ( i )-homology of $$THH(E_{2})$$ T H H ( E 2 ) using a Bökstedt spectral sequence and then lift the generating classes of K (1)-homology to fundamental classes in homotopy group of $$THH(E_{2})$$ T H H ( E 2 ) . These lifts allow us to construct terms of the cofiber sequence and explicitly understand how they map to $$THH(E_{2})$$ T H H ( E 2 ) .
We prove that, under a suitable condition on the contraction map of Hochschild homology, the shifted Hochschild cochain complex of a smooth proper stable ∞ -category is equivalent to an abelian dg Lie algebra. In particular, we show a generalization of Bogomolov–Tian–Todorov theorem to Calabi–Yau categories.
In this paper, we study the stabilization map on the relative Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal group. We obtain two bounds for the hyperbolic rank of the underlying quadratic space such that the stabilization map on K_1 - analogue of the relative DSER elementary orthogonal group becomes an isomorphism and a monomorphism.
We extend Wood's graph theoretic interpretation of certain quotients of the mod 2 dual Steenrod algebra to quotients of the mod p dual Steenrod algebra where p is an odd prime and to quotients of the C_2-equivariant dual Steenrod algebra. We establish connectedness criteria for graphs associated to monomials in these algebra quotients and investigate questions about trees and Hamilton cycles in these settings. We also give graph theoretic interpretations of algebraic structures such as the coproduct and antipode arising from the Hopf algebra structure on the mod p dual Steenrod algebra and the Hopf algebroid structure of the C_2-equivariant dual Steenrod algebra.
An isovariant map is an equivariant map between G-spaces which strictly preserves isotropy groups. In this paper, we lay the groundwork for the study of isovariant stable homotopy theory when G is a finite group. We prove an isovariant Blakers–Massey theorem and its n-cubical generalization, define a suitable notion of suspension (by a trivial representation sphere) in the isovariant category, and prove an isovariant Freudenthal suspension theorem.
We show that the relative cohomological dimension cd_R(G,H) of a relatively hyperbolic pair (G, H) is always finite when G does not contain R-torsion. We also show that this dimension is preserved under quasi-isometries, provided that G is torsion-free and the peripheral subgroup H is unconstricted and of type F_∞ . As a corollary of our methods, we compute cd_ℤ(G,H) in several cases.
We describe the main properties of the RO(C_2× Σ_2)-graded cohomology ring of a point and apply the results to compute the subring of motivic classes given by the Bredon motivic cohomology of real numbers and to compute RO(C_2× Σ_2)-graded cohomology ring of E_Σ_2C_2. This generalizes Voevodsky's identification of motivic cohomology of real numbers with the positive cone of RO(C_2) graded cohomology of a point.
We develop Weiss's manifold calculus in the setting of ∞-categories, where we allow the target ∞-category to be any ∞-category with small limits. We will establish the connection between polynomial functors, Kan extensions, and Weiss sheaves, and will classify homogeneous functors. We will also generalize Weiss and Boavida de Brito's theorem to functors taking values in arbitrary ∞-categories with small limits.
We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these fundamental groups are obtained as iterated semidirect products of subgroups of braid groups, with the structure for each component dictated by a finite rooted tree. These groups can be viewed as "braided" versions of the automorphism groups of such trees. We also discuss connections to statistical mechanics, topological data analysis, and geometric group theory.
In this article, we study the first, second, and third homology groups of the elementary group E_2(A) , where A is a commutative ring. In particular, we establish a refined Bloch–Wigner type exact sequence over a semilocal ring (subject to mild restrictions on its residue fields) in which either -1 ∈ (A^×)^2 or |A^×/(A^×)^2| ≤ 4 .
We show that for a large class of ∞-topoi there exist unstable arithmetic fracture squares, i.e. squares which recover a nilpotent sheaf F as the pullback of the rationalization of F with the product of the p-completions of F ranging over all primes p∈ℤ.
We generalize the concept of stack one dimension higher, introducing a notion of 2-stack suitable for a trihomomorphism from a 2-category equipped with a bitopology into the tricategory of bicategories. Moreover, we give a characterization of 2-stacks in terms of explicit conditions, that are easier to use in practice. These explicit conditions are effectiveness conditions for appropriate data of descent on objects, morphisms and 2-cells, generalizing the usual stacky gluing conditions one dimension higher. Furthermore, we prove some new results on bitopologies. The main one is that every object of a subcanonical bisite can be seen as the sigma-bicolimit of each covering bisieve over it. This generalizes one dimension higher a well-know result for subcanonical Grothendieck sites.
The Morava $E$-theories, $E_{n}$, are complex-oriented $2$-periodic ring spectra, with homotopy groups $\mathbb{W}_{\mathbb{F}_{p^{n}}}[[u_{1}, u_{2}, ... , u_{n-1}]][u,u^{-1}]$. Here $\mathbb{W}$ denotes the Witt vector ring. $E_{n}$ is a Landweber exact spectrum and hence uniquely determined by this ring as $BP_{\ast}$-algebra. Algebraic $K$-theory of $E_{n}$ is a key ingredient towards analyzing the layers in the $p$-complete Waldhausen $K$-theory chromatic tower. One hopes to use the machinery of trace methods to get results towards $K$-theory once the computation for $THH(E_{n})$ is known. In this paper we describe $THH(E_{2})$ as part of consecutive chain of cofiber sequences where each cofiber sits in the next cofiber sequence and the first term of each cofiber sequence is describable completely in terms of suspensions and localizations of $E_{2}$. For these results, we first calculate $K(i)$-homology of $THH(E_{2})$ using a B\"okstedt spectral sequence and then lift the generating classes of $K(1)$-homology to fundamental classes in homotopy group of $THH(E_{2})$. These lifts allow us to construct terms of the cofiber sequence and explicitly understand how they map to $THH(E_{2})$.
Motivated by the problem of constructing explicit geometric string structures, we give a rigid model for bundle 2-gerbes, and define connective structures thereon. This model is designed to make explicit calculations easier in applications to physics. To compare to the existing definition, we give a functorial construction of a bundle 2-gerbe as in the literature from our rigid model, including with connections. As an example we prove that the Chern–Simons bundle 2-gerbe from the literature, with its connective structure, can be rigidified—it arises, up to isomorphism in the strongest possible sense, from a rigid bundle 2-gerbe with connective structure via this construction. Further, our rigid version of 2-gerbe trivialisation (with connections) gives rise to trivialisations (with connections) of bundle 2-gerbes in the usual sense, and as such can be used to describe geometric string structures. The preprint of this article is available as arXiv:2209.05521 .
We study the rational homotopy theoretic and geometric properties of a construction which extends any cohomologically connected, finite type cdga to one satisfying cohomological Poincaré duality. Using this construction we show that non-trivial quadruple Massey products can pull back trivially under non-zero degree maps of Poincaré duality spaces, unlike the case of triple Massey products as studied by Taylor. We also show that a non-zero degree map between formal rational Poincaré duality spaces need not be formal. Our consideration of Massey products naturally ties in with cyclic A_∞ -algebras modelling Poincaré duality spaces.
Hypergraph is the most general model for complex networks involving group interactions. Taking the ideas of path homology from Alexander Grigor'yan, Yong Lin, Yuri Muranov and Shing-Tung Yau [18-22], Stephane Bressan, Jingyan Li and the authors of this article introduced embedded homology of hypergraphs [6] in 2019, which has leaded to successful applications in protein-ligand binding network [24, 25] in 2021. A fundamental question arising from practical applications is about the stability of the persistent embedded homology of hypergraphs. In this paper, we prove the stability of the persistent embedded homology as well as the persistent homology of the associated simplicial complex with respect to perturbations of the filtration on a hypergraph. We apply the persistent homology methods to morphisms of hypergraphs and prove the stability with respect to perturbations of the filtrations. We prove the constancy of the persistent Betti numbers under some conditions on the simple-homotopy types of hypergraphs.
In this paper, we introduce a simplicial analog of classifying spaces for commutativity which classify principal bundles with commutativity structure on their transition functions. Our construction W(τ,K), which takes as input a simplicial group K and a cosimplicial group τ that encodes the additional structure such as commutativity, is a variation of the W-construction for simplicial groups. Our main result shows that the geometric realization of our W(τ,K) is homotopy equivalent to the topological classifying space B(τ,|K|).
We compute the $$E_2$$ E 2 page of the Adams spectral sequence converging to the connective KO -theory of the second mod 2 Eilenberg–MacLane space, $$ko_*(K({\mathbb Z}_2,2))$$ k o ∗ ( K ( Z 2 , 2 ) ) , where $${\mathbb Z}_2$$ Z 2 is the cyclic group of order 2. This required a careful analysis of the structure of $$H^*(K({\mathbb Z}_2,2);{\mathbb Z}_2)$$ H ∗ ( K ( Z 2 , 2 ) ; Z 2 ) as a module over the subalgebra of the Steenrod algebra generated by $$\operatorname {Sq}^1$$ Sq 1 and $$\operatorname {Sq}^2$$ Sq 2 . Complete analysis of the spectral sequence is performed in [8].
