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In this paper, we introduce the signed barcode, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or, more generally, of a poset representation. Like its unsigned counterpart…

Given an acyclic map $X\to Y$ of closed manifolds dimension $d$, we study the relationship between the embeddings of $Y$ in $S^{n}$ with those of $X$ in $S^{n}$ when $n-d \ge 3$. The approach taken here is to first solve the Poincar\'e…

The Mapper algorithm is a popular tool for visualization and data exploration in topological data analysis. We investigate an inverse problem for the Mapper algorithm: Given a dataset $X$ and a graph $G$, does there exist a set of Mapper…

Let $G$ be a compact connected Lie group and $K$ a connected Lie subgroup. In this paper, we collect an assortment of results on equivariant formality of the isotropy action of $K$ on $G/K$. If the isotropy action of $K$ on $G/K$ is…

We establish various criteria for the inertness of the top cell attachments of Poincar\'{e} duality complexes through nonzero degree maps, algebraic intersection theory and various types of homotopy fibrations. Many examples are provided,…

In the past two decades, extensive research has been conducted on the (co)homology of various models of random simplicial complexes. So far, it has always been examined merely as a list of groups. This paper expands upon this by describing…

We give a variant of Naef's formula for the failure of invariance of the string topology coproduct under homotopy equivalences, using an obstruction class build from the higher homotopy data one can associate to a homotopy equivalence as…

In this article, we investigate properties of digital H-spaces in the graph theoretic model of digital topology. As in prior work, the results obtained often depend fundamentally on the choice between NP$_1$ and NP$_2$ product adjacencies.…

Let $X$ be an $(n-2)$-connected $2n$-dimensional Poincar\'e complex with torsion-free homology, where $n\geq 4$. We prove that $X$ can be decomposed into a connected sum of two Poincar\'e complexes: one being $(n-1)$-connected, while the…

We discuss $\pmb{SC}_*$, a simplicial homotopy model of $K(Z,2)$ constructed from circular permutations. In any dimension, the number of simplices in the model is finite. The complex $\pmb{SC}_*$ naturally manifests as a simplicial set…

We prove an asymptotic formula for the Euler characteristic of Kontsevich's commutative graph complex. This formula implies that the total amount of commutative graph homology grows super-exponentially with the rank and, via a theorem of…

Let TC$_n$(X) denote the n-th topological complexity of a topological space X. It is known that TC$_n$(X) does not exceed n-1 for non-contractible X, and so it makes sense to describe spaces X with TC$_n$(X) =n-1. Grant--Lupton--Oprea…

Karol Borsuk in 1968 asked: Is it true that every finite polyhedron dominates only finitely many different shapes? Danuta Kolodziejczyk showed that generally an answer to the Borsuk question is negative and also presented a positive answer…

Simplicial objects $\mathsf{sC}$ in descent categories $\mathsf{C}$, as introduced by Behrend and Getzler, provide a context in which to study higher stacks. In this note, we extend the construction of the canonical cocycle of a smooth…

Moss' theorem, which relates Massey products in the $E_r$-page of the classical Adams spectral sequence to Toda brackets of homotopy groups, is one of the main tools for calculating Adams differentials. Working in an arbitrary symmetric…

We discuss the algebra behind the matrix reduction algorithm for persistent homology, as presented in the paper ''Computing Persistent Homology'' by Afra Zomorodian and Gunnar Carlsson, in the lens of the more modern characterization of…

We prove the Gromov-Lawson-Rosenberg Conjecture for the group Z/4xZ/4 by computing the connective real k-homology of the classifying space with the Adams spectral sequence and two types of detection theorems for the kernel of the alpha…

A triangulation of a circle bundle $ E \xrightarrow[\text{}]{\pi} B$ is a triangulation of the total space $E$ and the base $B$ such that the projection $\pi$ is a simplicial map. In the paper we address the following questions: Which…

This survey provides an introduction to the Stolz-Teichner program on elliptic cohomology and quantum field theory.

In this paper, we construct a Real equivariant version of the B\"okstedt spectral sequence which takes inputs in the theory of Real Hochschild homology developed by Angelini-Knoll, Gerhardt, and Hill and converges to the equivariant…