# Questions tagged [gt.geometric-topology]

Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

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### The congruence subgroup property for mapping class groups and a conjecture of Grothendieck

This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them. Recall that a subgroup $K$ of a ...
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### Is there a Dehn-like presentation of a knot quandle?

The knot group can be presented using either a Wirtinger presentation (with generators corresponding to arcs of the knot diagram) or a Dehn presentation (with generators corresponding to regions of ...
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### What is the minimal genus of a surface acted on by the symmetric group $S_n$?

For $G$ a finite group, it is easy to construct a (connected, orientable) surface with a faithful action of $G$. E.g.: take a disjoint union of $G$ many spheres, and add a 1-handle for every edge in ...
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### Map from simplex to itself that preserves sub-simplices: revisited

Here it is proved that, if $f$ is a continuous map from an $n$-simplex $\Delta$ to itself, that maps each sub-simplex of $\Delta$ to itself, then $f$ must be onto $\Delta$ (surjective). I would like ...
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### Connectedness of Milnor fiber

Let $Q$ be a homogeneous polynomial in $n$ variables. Then it defines a locally trivial fiber bundle projection $Q:{\mathbb C}^n- Q^{-1}(0)\to {\mathbb C}-\{0\}$ (called Milnor fibration). Under what ...
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### Fractional Dehn Twist coefficient of monodromy of rational open book

Given an open book $(S,h)$, the fractional Dehn twist coefficient $c(h)$ in some sense measures the difference between $h$ and its Thurston representative $g$. More specifically, one can consider the ...
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### Intersection pairing on non-compact surface

Let $S$ be a smooth oriented connected $2$-manifold. We have an algebraic intersection pairing $\omega\colon H_1(S) \times H_1(S) \rightarrow \mathbb{Z}$. If $S$ is compact, then this is ...
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### "Almost embedding" the complete 2-dimensional complex $\mathcal K_7^2$ into $\Bbb R^4$

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 ...
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### Are finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ virtually special?

This might be a silly question--but are there any examples of finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ that are known to not be virtually special?
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### Extending diffeomorphisms between surfaces

Suppose we have two smooth compact oriented surfaces $M_1$ and $M_2$ with boundary,both of them have genus $g$, and there are orientation preserving diffeomorphisms $\psi_1, \psi_2, \cdots, \psi_n$, ...
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### Connected pre-images spanning $n$-cubes under dimension reducing maps
Let $I^n = [0,1]^n$ be the $n$-dimensional hypercube. For a continuous function $f: I^n \to \mathbb{R}^m$ with $m < n$, we're interested in the existence of points $p \in \mathbb{R}^m$ whose ...