代数拓扑

We prove that the random flag complex has a probability regime where the probability of nonvanishing homology is asymptotically bounded away from zero and away from one. Related to this main result we also establish new bounds on a sharp…

We prove that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented and that the rationalization map from this group to its rational analogue has a finite kernel.

We show that for any separably closed field $k$ of characteristic $p>0$, the canonical functor from nilpotent $p$-adic spaces to $\mathbb{E}_{\infty}$-coalgebras over $k$ (given by singular chains with coefficients in $k$) is fully…

We define $k$-rationalized $G$-equivariant elliptic cohomology, for a field of characteristic zero $k$ and a compact Lie group $G$, via adelic descent. We also give adelic descriptions of rationalized $G$-equivariant singular cohomology and…

Suppose $\Gamma$ is a finite group acting freely on $S^{n}$ ($n\geqslant 3$ being odd) and $M$ is any closed oriented $n$-manifold. We show that, given an integer $k$, the set $\deg^{-1}(k)$ of based homotopy classes of mappings with degree…

We prove that the formula for the diagonal approximation $\Delta_{K}$ on J. Stasheff's $n$-dimensional associahedron $K_{n+2}$ derived by the current authors in 2004 agrees with the "magical formula" for the diagonal approximation…

To any Adams-type spectrum $E$, Pstr\k{a}gowski produced a symmetric monoidal stable $\infty$-category $Syn_E$ whose objects are, in a sense, ''formal Adams spectral sequences''. $Syn_E$ comes equipped with a lax symmetric monoidal functor…

In this paper $Aut(\Sigma X\vee \Sigma Y)^\#$ the order of the group of self-homotopy equivalence of wedge spaces is studied. Under the condition of reducibility, we decompose $ Aut(\bigvee\limits_{t=1}^{k}X_{t})$ to the product of…

Let M be a simply-connected closed Poincare Duality complex of dimension n. Then M is obtained by attaching a cell of highest dimension to its (n-1)-skeleton M'. Conditions are given for when the skeletal inclusion i:M' --> M has the…

We define and develop a homotopy invariant notion for the sequential topological complexity of a map $f:X\to Y,$ denoted $TC_{r}(f)$, that interacts with $TC_{r}(X)$ and $TC_{r}(Y)$ in the same way Jamie Scott's topological complexity map…

Motivated by its link with functor homology, we study the prop freely generated by the operadic suspension of the operad Com. We exhibit a particular family of generators, for which the composition and the symmetric group actions admit…

We survey some results in the field of equivariant cobordism. In particular, we use methods from equivariant stable homotopy theory to calculate the unoriented $C_2$-equivariant bordism ring $\Omega^{C_2}_*$, which was originally calculated…

We show that the homology of the partition algebras, interpreted as appropriate Tor-groups, is isomorphic to that of the symmetric groups in a range of degrees that increases with the number of nodes. Furthermore, we show that when the…

We develop a generalisation of the path homology theory introduced by Grigor'yan, Lin, Muranov and Yau (GLMY-theory) in a general simplicial setting. The new theory includes as particular cases the GLMY-theory for path complexes and new…

We establish a general computational scheme designed for a systematic computation of characteristic classes of singular complex algebraic varieties that satisfy a Gysin axiom in a transverse setup. This scheme is explicitly geometric and of…

We calculate the rational cohomology of the classifying space of the diffeomorphism group of the manifolds $U_{g,1}^n:= \#^g(S^n \times S^{n+1})\setminus \mathrm{int}{D^{2n+1}}$, for large $g$ and $n$, up to approximately degree $n$. The…

We give a formula for the geometric fixed-points spectrum of the real topological cyclic homology of a bounded below ring spectrum, as an equaliser of two maps between tensor products of modules over the norm. We then use this formula to…

We introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects…

In this article, we compute the mod 2 representarion of the symmetric group of order 2 over the singular cohomology groups of orderd 2-configuration space $C_{2}(T^{d})$ of the $d$-torus $T^{d}$ for $d\geq 1$. As applications of the…

In this paper, we calculate the $n+3$, $n+4$ dimensional homotopy groups of indecomposable $\mathbf{A}_n^2$-complexes after localization at 2. This job is seen as a sequel to P.J. Hilton's work on the $n+1,n+2$ dimensional homotopy groups…