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Let $X$ be a 2-dimensional CW complex and let $c\subset X$ be a 2-cell attached along a simple closed loop $\gamma$ in the 1-skeleton $X^{(1)}$. For a natural number $n\ge 2$ consider the operation of …

A submanifold $X^n\subset Y^m$ is locally flat if each point $x\in X$ has a neighborhood $U\subset Y$ so that $(U,U\cap X)\simeq (\Bbb R^m, \Bbb R^n)$ with the standard embedding $\Bbb R^n\hookrightar …

A knot $K\subset\Bbb S^3=\partial \Bbb D^4$ is said to be (topolopgically) slice if there is a locally flat disk $D\subset\Bbb D^4$ with $\partial D=D\cap \Bbb S^3=K$. As far as I understand, locally …

Let $\mathcal K_7^2$ be the complete 2-dimensional simplicial complex on seven vertices, i.e. it has all $7\choose 2$ edges and all $7\choose 3$ 2-simplices (and no higher-dimensional simplices). I on …

Let $U\subseteq\Bbb R^n$ be an open set and $\gamma\subset U$ a closed null-homotopic curve in $U$ (i.e. it can be contracted to a point). Then is there an embedded disc $D\subset U$ with boundary $\g …

Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same bou …

Consider the following 2-dimensional CW-complex: its 1-skeleton is $K_8$, which we write as an edge-disjoint union $K_5\cup K_{5,3}\cup K_3$. Then for any two edges $ab\in E(K_5)$ and $cd\in E(K_3)$ t …

Let $D\subset\Bbb R^4$ be a PL-embedded 2-dimensional disc. Let $N=D+K$ be a thickening of the disc, where $K$ is some sufficiently small 4-dimensional PL-ball and "$+$" means Minkowski addition. Que …

I usually read about the Whitney trick in the context of smooth manifolds, but I wonder in how far it works in the piecewise linear (PL) category as well. I have a specific setting in mind that I will …

The paper Nonconvex Embeddings of the Exceptional Simplicial 3-Spheres with 8 Vertices by Mihalisin and Williams states this as a conjecture: Conjecture 5.2. …

Suppose I am given a smoothly slice knot $K\subset\Bbb S^3$. But I am only given a locally flat disc $D\subset \Bbb D^4$ with boundary $K$. Question: Is there a smooth disc $D'\subset\Bbb D^4$ with b …

I am given a 2-disc $D^2$ embedded into $\Bbb R^4$, that is, I have an injective continuous map $\phi:D^2\to\Bbb R^4$. I want to "double" this disc in the sense that I am looking for a second embedde …

Is there an embedding (i.e. injective continuous map) $$\phi:\Bbb R \Bbb P^2\hookrightarrow S^4\subseteq\Bbb R^5$$ of the projective plane $\Bbb R\Bbb P^2$ into the $4$-sphere, that is transitiv …