代数拓扑

In this paper, we describe the homotopy type of the homotopy fixed point sets of $S^3$-actions on rational spheres and complex projective spaces, and provide some properties of $S^1$-actions on a general rational complex.

Let $\rm{Aut}(p)$ denote the space of all self-fibre homotopy equivalences of a principal $G$-bundle $p: E\rightarrow X$ of simply connected CW complexes with $E$ finite. When $G$ is a compact connected topological group, we show that there…

We study degree one maps between closed orientable 4-manifolds with cyclic $\pi_1$, and obtain some results on the existence and finiteness, as well as some relation of $1$-domination and Euler Characteristics.

We prove that for any $\ell \geq 0$, there exists an algorithm which takes as input a description of a semi-algebraic subset $S \subset \mathbb{R}^k$ given by a quantifier-free first order formula $\phi$ in the language of the reals, and…

Let $n \geq 1$ be an integer, let $V=(\mathbb{Z}/2\mathbb{Z})^{n}$ and let $X$ be a $V$-CW-complex. If $X$ is a finite $CW$-complexe, the equivariant modulo $2$ cohomology of the $V$-CW-complexe $X$, denoted by $H_{V}^{*}(X,…

We study the cohomology of the free loop space of $SU(n+1)/T^n$, the simplest example of a complete flag manifolds and an important homogeneous space. Through this enhanced analysis we reveal rich new combinatorial structures arising in the…

The comparison map from bounded cohomology to singular cohomology plays an important role in the study of bounded cohomology theory and its applications. The vanishing and covering theorems of Gromov and Ivanov show interesting and useful…

As Reeb's theorem shows, Morse functions with exactly two singular points on closed manifolds are very simple and important. They characterize spheres whose dimensions are not $4$ topologically and the $4$-dimensional unit sphere. Special…

A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study various versions of Gromov's question and…

We study the Andr\'e-Quillen cohomology with coefficients of an algebra over an operad. Using resolutions of algebras coming from Koszul duality theory, we make this cohomology theory explicit and we give a Lie theoretic interpretation. For…

Let $X$ be a \text{\rm{2}}-connected and \text{\rm{6}}-dimensional CW-complex $X$ such that $H_{3}(X)\otimes\Z_2=0$. This paper aims to describe the group $\E(X)$ of the self-homotopy equivalences of $X$ modulo its normal subgroup…

We derive double coset formulae for the genus and extended genus of a finitely generated nilpotent group G, using the notions of bounded and bounded above automorphisms of $\prod G_S$, which are defined relative to a fixed fracture square…

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz \cite{F} and…

We compute the spectrum of the category of derived Mackey functors (in the sense of Kaledin) for all finite groups. We find that this space captures precisely the top and bottom layers (i.e. the height infinity and height zero parts) of the…

The rational homotopy type of a differential graded algebra (DGA) can be represented by a family of tensors on its cohomology, which constitute an $A_\infty$-minimal model of this DGA. When only the cohomology is needed to determine the…

We prove a filtered version of the Homotopy Transfer Theorem which gives an A-infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the…

For a finitely presented discrete group $\Gamma$, we introduce two generalizations of the orbifold Euler characteristic and $\Gamma$-orbifold Euler characteristic to a class of proper topological groupoids large enough to include all…

The $2$-primary homotopy $\beta$-family, defined as the collection of Mahowald invariants of Mahowald invariants of $2^i$, $i \geq 1$, is an infinite collection of periodic elements in the stable homotopy groups of spheres. In this paper,…

Let $f:G\to \mathrm{Pic}(R)$ be a map of $E_\infty$-groups, where $\mathrm{Pic}(R)$ denotes the Picard space of an $E_\infty$-ring spectrum $R$. We determine the tensor $X\otimes_R Mf$ of the Thom $E_\infty$-$R$-algebra $Mf$ with a space…

We study the Morava $E$-theory (at a prime $p$) of $BGL_d(F)$, where $F$ is a finite field with $|F|=1\pmod{p}$. Taking all $d$ together, we obtain a structure with two products $\times$ and $\bullet$. We prove that it is a polynomial ring…