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We study several different notions of algebraicity in use in stable homotopy theory and prove implications between them. The relationships between the different meanings of algebraic are unexpectedly subtle, and we illustrate this with…
The mod 2 Moore spectrum $C(2)$ is the cofiber of the self-map $2: \mathbb{S} \to \mathbb{S}$. Building on work of Burklund, Hahn, and Senger, we prove that above a line of slope $\frac{1}{5}$, the Adams spectral sequence for $C(2)$…
Beben and Theriault proved a theorem on the homotopy fiber of an extension of a map with respect to a cone attachment, which has produced several applications. We give a short and elementary proof of this theorem.
In the late 80s, V.~Arnold and V.~Vassiliev initiated the topological study of the space of real univariate polynomials of a given degree d and with no real roots of multiplicity exceeding a given positive integer. Expanding their studies,…
Let $M$ and $N$ be fiber bundles over the same base $B$, where $M$ is endowed with a free involution $\tau$ over $B$. A homotopy class $\delta \in [M,N]_{B}$ (over $B$) is said to have the Borsuk-Ulam property with respect to $\tau$ if for…
We study the homotopy decompositions of the suspension $\Sigma M$ of an $(n-1)$-connected $(2n+2)$ dimensional Poincar\'{e} duality complex $M$, $n\geq 2$. In particular, we completely determine the homotopy types of $\Sigma M$ of a…
Autonomous motion of a system (robot) is controlled by a motion planning algorithm. A sequential parametrized motion planning algorithm \cite{FP22} works under variable external conditions and generates continuous motions of the system to…
We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients, of $CP_{C_2}^\infty = B_{C_2} U(1)$, the complex projective space, a model for the classifying space for $C_2$-equivariant complex line bundles. The…
In a first time we present a version of the Poincar{\'e}-Lefschetz theorem for certain cellular cosheaves on a particular subdivision of a CW-complex K. To that end we construct a cellular sheaf on K whose cohomology with compact support is…
Let $G$ be a Lie group with a maximal torus $T$. Combining Schubert calculus in the flag manifold $G/T$ with the Serre spectral sequence of the fibration $G\rightarrow G/T$, we construct the integral cohomology ring $H^{\ast}(G)$ uniformly…
Symmetry is ubiquitous throughout nature and can often give great insights into the formation, structure and stability of objects studied by mathematicians, physicists, chemists and biologists. However, perfect symmetry occurs rarely so…
The third string bordism group $\mathrm{Bord}_3^{\mathrm{String}}$ is known to be $\mathbb{Z}/24\mathbb{Z}$. Using Waldorf's notion of a geometric string structure on a manifold, Bunke--Naumann and Redden have exhibited integral formulas…
We introduce a persistent Hochschild homology framework for directed graphs. Hochschild homology groups of (path algebras of) directed graphs vanish in degree $i\geq 2$. To extend them to higher degrees, we introduce the notion of…
\"Uberhomology is a recently defined homology theory for simplicial complexes, which yields subtle information on graphs. We prove that bold homology, a certain specialisation of \"uberhomology, is related to dominating sets in graphs. To…
Jang has proven a remarkable classification of $6$-dimensional manifolds having an almost complex circle action with $4$ fixed points. Jang classifies the weights and associated multigraph into six cases, leaving the existence of connected…
The paper is devoted to one of the important notions of the shape theory: that of strong movability, which was primarily introduced by K. Borsuk for metrizable compacts. A strong movability criterion is proved for topological spaces, which…
The integral cohomology algebra of $\widetilde G_{6,3}$ has been determined in the recent work of Kalafat and Yal\c{c}inkaya. We completely determine the integral cohomology algebra of $\widetilde G_{n,3}$ for $n=8$ and $n=10$. The main…
I introduce the concept of a persistence diagram (PD) bundle, which is the space of PDs for a fibered filtration function (a set $\{f_p: \mathcal{K}^p \to \mathbb{R}\}_{p \in B}$ of filtrations that is parameterized by a topological space…
Clustering is one of the most common tasks of Machine Learning. In this paper we examine how ideas from topology can be used to improve clustering techniques.
This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective `proper' alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from…