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Results tagged with differential-topology
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user 89741
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
8
votes
2
answers
384
views
Open book decompositions in dimension 4
The question about the existence of open book decompositions for a closed oriented $n$-dimensional manifold seems to be answered in all dimensions except dimension $4$, where as far as I can tell the …
- 1,174
4
votes
1
answer
401
views
Paracompactness of Quotient by Group Action
Suppose $X$ is a metric space with a free group action by a topological group $G$, which is also a metric space, such that $\pi\colon X \to X/G$ is a fiber bundle.
Does the quotient inherit the para …
- 1,174
11
votes
2
answers
668
views
Can we embed a closed manifold into a homotopy equivalent CW complex?
Suppose $X$ is a CW complex and $M$ is a closed manifold and suppose further that there exists a homotopy equivalence $X \simeq M$. Does there exists an embedding of $M$ into $X$ (i.e. an injective (p …
- 1,174
2
votes
1
answer
118
views
Hyperbolization with word-hyperbolic fundamental group
In Davis-Januszkiewica´s paper Hyperbolization of polyhedra it is shown that for every manifold $M$ there exists a map $N \to M$ of non-zero degree such that $N$ is aspherical (plus some more properti …
- 1,174
6
votes
Can we embed a closed manifold into a homotopy equivalent CW complex?
While thinking about it with a friend, we came up with the following two dimensional counter example:
Take the standard knot diagram of the trefoil knot (as a self-intersecting curve in $\mathbb{R}^2$ …
- 1,174
8
votes
0
answers
219
views
Representing the fundamental class of an aspherical manifold in the bar complex
Suppose $M$ is a compact orientable aspherical manifold and $G$ its fundamental group. Is there a nice description of representatives of the fundamental class of $M$ and its dual in the (homogenous) b …
- 1,174
8
votes
Parallelizability of 3-manifolds
Suppose we have a non-compact $3$ dimensional manifold $M$ which is spinable. Fix a CW structure of the manifold, then being spinable is equivalent to the tangent bundle being trivial on the $2$-skele …
- 1,174
7
votes
1
answer
196
views
Lipschitz bounds and homotopy groups of diffeomorphism groups
Let $M$ denote a closed Riemannian manifold. Let $\mathrm{Diff}_0^L(M)$ denote the supspace of the identity component of the diffeomorphism group $\mathrm{Diff}_0(M)$ of diffeomorphisms with Lipschitz …
- 1,174
9
votes
1
answer
434
views
Action of diffeomorphism group on non-vanishing vector fields
Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0 …
- 1,174
10
votes
3
answers
656
views
Doubles of 2-handlebodies
Let $X$ denote a $4$-manifold with boundary obtained by adding $k_1$ $1$-handles to $B^4$ and $k_2$ many $2$-handles to the resulting manifold i.e. $X$ is an arbitrary $4$-dimensional $2$-handlebody. …
- 1,174
4
votes
Accepted
Doubles of 2-handlebodies
Let $X$ denote a $2$-handlebody. I claim that the inclusion $\partial X \to X$ induces a surjection on fundamental groups. Indeed, let $Y\subset X$ denote the underlying $1$-handlebody (i.e. the union …
- 1,174
12
votes
1
answer
368
views
Fundamental group of the complement of a codimension two submanifold
Let $M$ denote an arbitrary closed, connected, n-dimensional manifold for $n\geq 4$. Does there always exist a closed (not necessarily connected!) codimension two submanifold $S \subset M$ such that $ …
- 1,174