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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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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When can we extend a diffeomorphism from a surface to its neighborhood as identity?
Let $M$ be a closed and simply-connected 4-manifold and let $f: M^4 \to M^4$ be a diffeomorphism such that $f^*: H^*(M;\mathbb{Z})\to H^*(M;\mathbb{Z})$ is the identity map. Moreover, let $\Sigma \...
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Low dimensional homotopy groups of $\operatorname{Top}(4)$
$\DeclareMathOperator\Top{Top}$I would like to compute $\pi_3\Top(4)$ and $\pi_4\Top(4)$. It is known that $\Top(4)/O(4) \rightarrow \Top/O$ is 5-connected and
$$
\pi_k(\Top/O) =
\begin{cases}
...
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Sheaf cohomology of non-paracompact manifolds (e.g. the long line)
I have long heard that manifolds are "affine". If we allow non-paracompact manifolds, then this seems to fail, since as explained in Dmitri Pavlov's answer, the Serre–Swan theorem fails. I ...
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Does there exist a Dehn filling of an irreducible 3-manifold with toroidal boundaries which is still irreducible?
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible toroidal boundary (there might be more than one boundary component). Is it always possible to choose appropriate slopes on ...
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Lengths of generators of surface group
Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some ...
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Property P and R for general 3-manifolds
Let $Y$ be a closed oriented $3$-manifold and $K$ be a knot in $Y$. We say $K$ is the unknot if $K$ is contained in a local $3$-ball in $Y$ and is unknotted therein.
Generalized Property R:
If a Dehn ...
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Generalized Stokes's theorem and the relationship of volume to surface area of objects of arbitrary genus
In this post I saw that it could be explained with the Generalized Stokes's theorem why the derivative of the area of a circle is equal to the boundary of the circle (the circumference):
$$C=\frac{d}{...
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Must the number of smooth structures be countable or continuum?
Let $M$ be a manifold. Must the number of non-diffeomorphic smooth structures on $M$ be either countable or continuum? Could it be something in between when the continuum hypothesis fails?
Edit:
By ...
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Distributions of random walks on boundaries of balls in hyperbolic metric spaces
Suppose $G$ is a finitely-generated non-elementary hyperbolic group and consider a symmetric random walk on the Cayley graph $\text{Cay}(G,S)$ with generating set $S$. Denote the set of points $B_{\...
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