All Questions
Tagged with gt.geometric-topology dg.differential-geometry
508 questions
3
votes
1
answer
269
views
$\mathrm{SL}(2,\mathbb{Z})$ finitely generated by using the Schwarz-Milnor lemma
Recently, I have been studying the modular group $G=\mathrm{SL}(2,\mathbb{Z})$, and I am trying to prove $G$ is finitely generated by using the Schwarz-Milnor lemma in geometric group theory.I am ...
- 111
2
votes
1
answer
380
views
Is it always possible to connect the endpoints of a smooth injective path, so the resulting path is smooth, closed and simple?
Motivation
The motivation for this question arises from the following problem: Is there a closed, simple, and infinitely differentiable path on the (complex) plane such that every straight line has a ...
4
votes
0
answers
177
views
Basis of topology on space of properly embedded smooth manifolds
In A Short Exposition of the Madsen-Weiss Theorem, Hatcher discusses (starting at p.6) a basis for a topology on the space $\mathcal{C}^n$, the space of all smooth oriented $d$-dimensional ...
- 141
2
votes
0
answers
82
views
Is isoperimetric hypersurface unique up to homeomorphism?
Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following:
$$\...
- 356
2
votes
0
answers
116
views
Generalized Stokes's theorem and the relationship of volume to surface area of objects of arbitrary genus
In this post I saw that it could be explained with the Generalized Stokes's theorem why the derivative of the area of a circle is equal to the boundary of the circle (the circumference):
$$C=\frac{d}{...
- 131
7
votes
2
answers
614
views
Locally conformally flat
Is there any example of a locally conformally flat manifold that is neither a space form nor a product of space forms?
- 71
3
votes
1
answer
529
views
Zeros of a function defined on $\mathbb{S}^2 \times \mathbb{S}^2$
Let $u$ be a smooth function on the sphere, and for each $y \in \mathbb{S}^2$, let $R_y$ be the $180^\circ$ rotation about the vector $y$. For each pair $(x, y) \in \mathbb{S}^2 \times \mathbb{S}^2$, ...
- 434
1
vote
0
answers
107
views
Extend a circle action on $3$-manifolds
Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action.
Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, ...
- 891
3
votes
2
answers
265
views
For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?
Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra ...
- 1,467
0
votes
0
answers
77
views
Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations
Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
- 383
3
votes
1
answer
200
views
Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?
It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
- 491
1
vote
1
answer
228
views
Submersion of real projective space into Euclidean space
The famous Whitney immersion theorem states that any real projective space $\mathbb{RP}^n$ can be immersed into $\mathbb{R}^{2n}$.
However, I haven't found information about the submersion ...
- 173
5
votes
1
answer
415
views
A question about the existence of spin maps
Let $M, N$ be two smooth manifolds, not necessarily spin. My question is the following:
How can we construct a non-constant spin map $f:M\to N$ of degree zero?
Here spin map means that $f$ preserves ...
- 563
0
votes
1
answer
376
views
Relation between trivial tangent bundle $\Leftrightarrow$ certain characteristic classes of tangent bundle vanish [closed]
We know that
framing structure means the trivialization of tangent bundle of manifold $M$.
string structure means the trivialization of Stiefel-Whitney class $w_1$, $w_2$ and half of the first ...
- 447
4
votes
2
answers
583
views
Compactification of a product of manifolds
Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...
- 319
1
vote
1
answer
192
views
Lie group framing and framed bordism
What is the definition of Lie group framing, in simple terms?
Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
- 447
2
votes
0
answers
175
views
Minimal first Pontryagin class $p_1=1$?
From Hirzbuch theorem,
the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$.
I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$.
Is ...
- 447
2
votes
1
answer
165
views
string bordism group and framed bordism group for $d \leq 6$ and $d \geq 7$
Why do the string bordism group and the framed bordism group
coincide the same in dimensions lower than 7 ($d = 0,1,2,3,4,5, 6$)?
Why do the string bordism group and the framed bordism group differ
...
- 10.5k
16
votes
0
answers
425
views
Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
- 587
21
votes
1
answer
2k
views
Intuition behind manifolds which are homeomorphic but not diffeomorphic
Popular articles on mathematics often explain the difference between homeomorphism and diffeomorphism with statements like - "A rectangle is homeomorphic to the circle but not diffeomorphic to it&...
- 463
0
votes
0
answers
329
views
Pushforwards in vector bundles over a topological spaces
I have been reading the discussion from Pushforward and pullback..
I understand that it is quite straight forward to construct a pullback of a vector bundle. In the discussion it is clear that if we ...
- 615
6
votes
1
answer
248
views
How small need a perturbation be to not change the diffeomorphism type of a variety?
Let $f,g \in \mathbb{R}[x_0,\dots,x_k]$ be homogeneous polynomials and $X:=Z(f) \subset \mathbb{RP}^k$ be the projective variety defined by $f$.
Assume that $X$ is smooth and has codimension $1$.
Then ...
- 577
10
votes
2
answers
1k
views
Exotic smooth structure
M.Freedman and R.Gompf's work show that there are at least 13 exotic structures in $S^3\times \mathbb{R}$, which is a open 4-manifold, so now I wonder whether there is an exotic structure in $S^3\...
- 101
2
votes
0
answers
127
views
Foliation of $X$ by once punctured planes without any singularities
Let $n=3.$
Take $X=(0,1)^n.$ Fix points $p,q$ s.t. $\text{dist}_n(p,q)=\sqrt{n}.$ Construct a smooth regular foliation of $X$ with $(n-1)-$dim. leaves which are topologically $(0,\sqrt{n})\times S^{n-...
- 383
3
votes
0
answers
75
views
non-negative curvature condition for polyhedral manifolds
A polyhedral manifold P, i.e, a topological manifold with a triangulation where each simplex is isometric to a simplex in Euclidean space (other constant curvature spaces are allowed), is said to have ...
- 85
12
votes
1
answer
623
views
Definition of Thurston's skinning map
A key construction in Thurston's proof of the existence of hyperbolic structures on Haken manifolds is the so-called "skinning map" associated to a 3-manifold $M$ with boundary whose ...
- 197
1
vote
0
answers
130
views
Instantons on the 4-sphere with respect to other Riemannian metrics
It is known that the moduli space of $\text{SU}(2)$ instantons of charge $1$ on $S^4$ is diffeomorphic to the five-ball $B^5$ if $S^4$ is endowed with the round metric.
Question: what does the moduli ...
- 915
5
votes
1
answer
368
views
Six people standing on earth
Consider 6 people $p_i$, $i=1,\dots 6$, standing on a sphere $S^2$. We label the positions of these people by $p_i$ again. Suppose no pair of these points $p_i$ are antipodal. At each point $p_i$ ...
- 434
3
votes
1
answer
175
views
Example of homeomorphism that lifts to real blow up but not C^1?
Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(...
- 41
5
votes
0
answers
289
views
A certain kind of proof of the Hairy Ball Theorem
I'd just like to know if proofs of the Hairy Ball Theorem along the following lines are well-known or even somewhere in the literature.
From a given vector field $V_1$ on $S^2$, form another, $V_2$, ...
- 17.6k
1
vote
0
answers
97
views
about codimension two foliation
Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold
I am curious about examples of codimension
Are there any previous studies or lecture notes of foliation ...
- 73
17
votes
1
answer
506
views
Topology of the space of embedded genus $g$ surfaces in $S^3$
Consider the space of smoothly embedded genus $g$ surfaces in the 3-sphere under the $C^\infty$ topology:
$$\mathcal E_g:=\operatorname{Emb}(\Sigma_g,S^3)/{\operatorname{Diff}(\Sigma_g)}$$
where $\...
- 525
4
votes
0
answers
113
views
topological and $\mathscr{C}^{\infty}$-circle bundles
Let $M$ be a closed $\mathscr{C}^{\infty}$-manifold. Suppose that the underlying topological space of $M$ has a topological circle bundle structure $S^1\hookrightarrow M\to B$. Does $M$ admit a $\...
- 113
2
votes
1
answer
190
views
Estimating the volume of a convex shape in higher dimensions based only on normal sections
We are given a $d$-dimensional convex shape $S$ inscribed in the hypercube $[-1,1]^d$. We want find an approximation of its volume based only on a set of curves given by the intersection of the $S$ ...
- 1,677
3
votes
0
answers
608
views
Show that continuous maps between smooth manifolds can be approximated by smooth maps WITHOUT using Whitney's embedding theorem
As it is well-known (and for example this question shows) each continuous map between smooth manifolds is homotopically equivalent to a smooth map that can be constructed using the Whitney embedding ...
- 1,149
3
votes
0
answers
195
views
Is there such an isotopy for every homology sphere?
Let $n \geq 3$, and $\Sigma^{n-1} \subset \mathbf{S}^n$ be a smoothly and properly embedded, orientable, and connected submanifold of the sphere. This divides the sphere into two open sets, $U_-$ and $...
- 5,038
3
votes
0
answers
150
views
A question about index of Dirac operator
Let $\Phi: M\to S^n$ be a map from an even-dimensional, $\dim M=n$, spin manifold $M$ with the boundary $\partial M$ to a unit sphere. And $\Phi$ is locally constant near $\partial M$. If we take a ...
- 563
3
votes
1
answer
239
views
Does the Weeks manifold have the smallest volume among all finite volume oriented hyperbolic 3-manifolds?
It is a result of Chinburg-Friedman-Jones-Reid that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold.
There is also a result of Milley that says that if $N$ is a closed ...
- 33
4
votes
0
answers
230
views
Trivial fibration over $S^1$ and closed 1-form
Tischler's theorem says that a closed differentiable manifold $M$ has a nondegenerate real closed 1-form if and only if $M$ is a fiber bundle over the circle $S^1$.
Except for the condition "...
- 41
3
votes
1
answer
244
views
Partitioning a smooth manifold into geodesically convex sets
Let $X$ be a connected and compact $d$-dimensional smooth manifold; where $d$ is a positive integer. Does (or rather, when does) there exist a metric $\rho$ on $X$ generating $X$'s topology and a ...
- 5,405
0
votes
0
answers
56
views
Embedding and compactness of thickened family of graphs
Let $\{G_n\}_\mathbb{N}$ be a family of graphs $G_n=(V_n,E_n)$ for which $|V_n|$ and $|E_n|$ tend to infinity. I would like to know if the family of objects $\{M_n\}_\mathbb{N}$ obtained by thickening ...
1
vote
2
answers
148
views
Construct a hypersurface with fixed principal curvatures at a point
I'm reading Eschenburg's paper Local convexity and nonnegative curvature —
Gromov's proof of the sphere theorem recently. And I meet a little question: Given a point $p\in M$, $N\in T_pM$, we want to ...
- 241
8
votes
2
answers
458
views
Examples of homology sphere that bound a nonsmoothable contractible 4-manifold
Freedman’s theorem shows that all 3-dimensional homology spheres bound topologically a contractible 4-manifold. It is well known that the Poincaré homology sphere does not bound a smooth contractible ...
- 2,623
2
votes
1
answer
119
views
Density of smooth bi-Lipschitz maps in smooth maps
Setup/Motivation:
Let $(M,g)$ and $(N,\rho)$ be complete Riemannian manifolds of respective dimensions $m$ and $n$ and suppose that $m\leq n$. Let $\operatorname{bi-C}^{\infty}(M,N)$ denote the class ...
- 195
8
votes
0
answers
179
views
Sensitivity of topological field theories
I am struggling to find references or studies that explore the overall sensitivity of topological field theories as an invariant of smooth manifolds. There is the paper by Davis that explores how ...
- 331
2
votes
0
answers
117
views
About connected cobordism and surgery
I need to find various ways of performing two surgeries on a collection of circles so that the resulting 2-dimensional cobordism (the trace of the surgeries) is connected.
How can I find these ? up ...
- 119
0
votes
1
answer
222
views
Why do we have sixteen possible configurations of three saddles on one level?
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030) page 13 we have :
There are sixteen ...
- 119
2
votes
1
answer
311
views
Vanishing cycles exact sequence for degeneration of curves
Let $f : X\rightarrow D$ be a family of smooth projective curves over the complex unit disk $D$, degenerating to a nodal curve $X_0$ above $0\in D$.
Let $\eta\in D - \{0\}$ be a general point, and let ...
- 7,527
2
votes
1
answer
130
views
Gluing isotopic smoothings
Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the ...
- 803
1
vote
0
answers
157
views
Decomposing the homology of a connected sum of surfaces in a way which highlights the combinatorics of gluing
For each $i = 1,2$ and $j = 1,\ldots,n$, let $C_{i,j}$ be a connected compact oriented surface with $k$ boundary components. For $i = 1,2$, let $C_i = \sqcup_{j=1}^n C_{i,j}$, and let $C$ be the ...
- 7,527