The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that $\pi_0$ of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable $\mathbb{C}$-valued $\mathbb{Z}$-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators.
We establish that the category of Silva spaces, aka $\mathrm{LS}$-spaces, formed by countable inductive limits of Banach spaces with compact linking maps as objects and linear and continuous maps as morphisms, is not an integral category. The result carries over to the category of $\mathrm{PLS}$-spaces, i.e., countable projective limits of $\mathrm{LS}$-spaces—which contains prominent spaces of analysis such as the space of distributions and the space of real analytic functions. As a consequence, we obtain that both categories neither have enough projective nor enough injective objects. All results hold true when ‘compact’ is replaced by ‘weakly compact’ or ‘nuclear’. This leads to the categories of $\mathrm{PLS}$-, $\mathrm{PLS_w}$- and $\mathrm{PLN}$-spaces, which are examples of ‘inflation exact categories with admissible cokernels’ as recently introduced by Henrard, Kvamme, van Roosmalen and the second-named author.
This paper generalizes a result of Lynn on the “degree” of an equivariant cohomology ring $H^\ast_G (X)$. The degree of a graded module is a certain coefficient of its Poincaré series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree:
We also show how this formula relates to the additivity formula from commutative algebra, demonstrating both the algebraic and geometric character of the degree invariant.
Recently, Asao and Izumihara introduced CW-complexes whose homology groups are isomorphic to direct summands of the graph magnitude homology group. In this paper, we study the homotopy type of the CW-complexes in connection with the diagonality of magnitude homology groups. We prove that the Asao–Izumihara complex is homotopy equivalent to a wedge of spheres for pawful graphs introduced by Y. Gu. The result can be considered as a homotopy type version of Gu’s result. We also formulate a slight generalization of the notion of pawful graphs and find new non-pawful diagonal graphs of diameter $2$.
Let $L$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_L$. We show that periodic topological cyclic homology of $\mathcal{O}_L$, over the base $\mathbb{E}_\infty$-ring $\mathbb{S}_{W(\mathbb{F}_q)} [z]$ carries a $p$-height one formal group law $\operatorname{mod}(p)$ that depends on an Eisenstein polynomial of $L$ over $\mathbb{Q}_p$ for a choice of uniformizer $\varpi \in \mathcal{O}_L$.
We provide a new description of the complex computing the Hochschild homology of an $H$-unitary $A_\infty$-algebra $A$ as a derived tensor product $A \oplus^\infty_{A^\epsilon}$ such that: (1) there is a canonical morphism from it to the complex computing the cyclic homology of $A$ that was introduced by Kontsevich and Soibelman, (2) this morphism induces the map $I$ in the well-known SBI sequence, and (3)$H^0 \left( (A \oplus^\infty_{A^\epsilon} A)^\# \right)$ is canonically isomorphic to the space of morphisms from $A$ to $A^\#$ in the derived category of $A_\infty$-bimodules. As direct consequences we obtain previous results of Cho and Cho–Lee, as well as the fact that Koszul duality establishes a bijection between (resp., almost exact) $d$-Calabi–Yau structures and (resp., strong) homotopy inner products, extending a result proved by Van den Bergh.
The self-closeness number of a CW-complex is a homotopy invariant defined by the minimal number $n$ such that every selfmap of $X$ which induces automorphisms on the first $n$ homotopy groups of $X$ is a homotopy equivalence. In this article we study the self-closeness numbers of finite Cartesian products, and prove that under certain conditions (called reducibility), the self-closeness number of product spaces is equal to the maximum of the self-closeness numbers of the factors. A series of criteria for the reducibility are investigated, and the results are used to determine self-closeness numbers of product spaces of some special spaces, such as Moore spaces, Eilenberg–MacLane spaces or atomic spaces.
We show that in a fibration the coformality of the base space implies the coformality of the total space under reasonable conditions, and these conditions can not be weakened. The result is partially dual to the classical work of Lupton on the formality within a fibration. Our result has two applications. Firstly, we show that for certain homotopy cofibrations, the coformality of the cofiber implies the coformality of the middle space. Secondly, we show that the total spaces of certain spherical fibrations are Koszul.
Let $n \gt 2$ and $\mathcal{G}_k (\mathbb{C}P^2)$ be the gauge groups of the principal $Sp(n)$-bundles over $\mathbb{C}P^2$. In this article we partially classify the homotopy types of $\mathcal{G}_k (\mathbb{C}P^2)$ by showing that if there is a homotopy equivalence $\mathcal{G}_k (\mathbb{C}P^2) \simeq \mathcal{G}_{k^\prime} (\mathbb{C}P^2)$ then $(k, 4n(2n + 1)) = (k^\prime , 4n(2n + 1))$.