All Questions
149 questions
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
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
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
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
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
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
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)}$$
where $\...
- 525
3
votes
0
answers
195
views
Is there such an isotopy for every homology sphere?
Let $n \geq 3$, and $\Sigma^{n-1} \subset \mathbf{S}^n$ be a smoothly and properly embedded, orientable, and connected submanifold of the sphere. This divides the sphere into two open sets, $U_-$ and $...
- 5,038
2
votes
1
answer
311
views
Vanishing cycles exact sequence for degeneration of curves
Let $f : X\rightarrow D$ be a family of smooth projective curves over the complex unit disk $D$, degenerating to a nodal curve $X_0$ above $0\in D$.
Let $\eta\in D - \{0\}$ be a general point, and let ...
- 7,527
2
votes
1
answer
130
views
Gluing isotopic smoothings
Let $M$ be a topological manifold which can be written as $M = U \cup V$ where $U$ and $V$ are open. Suppose both $U$ and $V$ admit smooth structures. Also assume that on the overlap $U \cap V$ the ...
- 803
1
vote
0
answers
157
views
Decomposing the homology of a connected sum of surfaces in a way which highlights the combinatorics of gluing
For each $i = 1,2$ and $j = 1,\ldots,n$, let $C_{i,j}$ be a connected compact oriented surface with $k$ boundary components. For $i = 1,2$, let $C_i = \sqcup_{j=1}^n C_{i,j}$, and let $C$ be the ...
- 7,527
3
votes
0
answers
137
views
Intersection number for 4 manifold with boundary
Let $X$ be a closed oriented smooth $4$-manifold. Suppose there is an embedding $\Sigma\to X$, it is known that the self-intersection number satisfies $[\Sigma]\cdot [\Sigma]=\pm\int_\Sigma c_1(N)$, ...
- 1,915
6
votes
1
answer
644
views
Torus bundles and compact solvmanifolds
I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.
Let
$$
T^n \to M \to T^m ...
- 4,081
5
votes
0
answers
132
views
geometry and connected sum of aspherical closed manifolds
Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then
$$
\Gamma \backslash G/K
$$
is an aspherical ...
- 4,081
1
vote
0
answers
284
views
A question on existence of gradient vector field on manifold with boundary
Let $M$ be a compact manifold with smooth boundary $\partial M$. Does $M$ admit a gradient vector field $\nabla u$, which has no zeros, i.e. $\nabla u(x)\neq 0$, $\forall x\in M\cup\partial M$?
Thanks ...
- 51
2
votes
1
answer
484
views
Mapping torus of orientation reversing isometry of the sphere
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$
Let $ f_n $ be an orientation reversing isometry of the round ...
- 4,081
17
votes
1
answer
898
views
Does there exist a closed manifold with vanishing reduced rational cohomology but nonvanishing odd torsion cohomology?
Question: Let $p$ be an odd prime. Does there exist a closed manifold $M$ with $\widetilde H^\ast(M; \mathbb Q) = 0$ but $\widetilde H^\ast(M; \mathbb F_p) \neq 0$?
When $p = 2$, an example is given ...
- 63.9k
0
votes
0
answers
266
views
Define a characteristic class on a simplicial complex (non-manifold)
Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class?
(Please provide Yes or No answers, and reasonings.)
Given a fixed ...
- 10.5k
7
votes
1
answer
260
views
Bordism for oriented triangulable manifolds without smooth differentiable structures
We know the bordism group for oriented smooth differentiable structures such as $\Omega_d^{SO}$ that requires the special orthogonal group structure on the tangent bundle $TM$ of manifold $M$.
$$\...
- 10.5k
0
votes
1
answer
154
views
Why does $X_0\times S^1\simeq X-X_0$? [closed]
Let $X$ be an $n$-dimensional connected smooth manifold, and let $X_0$ be an embedded $(n-2)$-dimensional compact submanifold of $X$ with the trivial normal bundle. How do we get inclusion?
$$X_0\...
- 563
4
votes
1
answer
381
views
Injectivity of map of fundamental groups from totally geodesic hypersurface
Let $X$ be a compact manifold of non-positive sectional curvature which carries a connected totally geodesic hypersurface $X_0\subset X$. Let $K$ be any compact subset of $X-X_0$. That's to say we ...
- 563
5
votes
1
answer
371
views
Orientation reversal and restriction to submanifold of lower dimension
Let $M$ be a connected closed oriented manifold with at least one orientation-reversing homeomorphism $M\to M$.
Let $S\subset M$ be a connected closed embedded submanifold of lower dimension. Let $f:M\...
- 59
1
vote
0
answers
151
views
Density of $G$-invariant morse functions
Let $G$ be a finite group acting on a compact manifold $M$. Let $f$ be a $G$-invariant smooth function. Can it be approximated by $G$-invariant Morse functions?
- 99
4
votes
0
answers
158
views
Postnikov square explicitly on a simplicial complex
$\DeclareMathOperator\Z{\mathbb{Z}}$
Following Wikipedia, a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, introduced by ...
- 10.5k
7
votes
0
answers
350
views
Do smooth maps with nowhere-maximal rank have small image?
I’m trying to better understand the concept of “maps with small image” as used by Lipyanskiy in his construction of “geometric homology” in https://arxiv.org/abs/1409.1121. Lipyanskiy utilizes ...
- 5,534
3
votes
0
answers
194
views
The first Stiefel Whitney class v.s. fermion eta invariant v.s. spin structure v.s. $H^1(M,\mathbb{Z}_2)$
$H^1(M,\mathbb{Z}_2)$ specifies the 1st cohomology class of manifold $M$ (can be regarded as spacetime) with $\mathbb{Z}_2$ coefficient,
it is often to see that we say the 1st Stiefel Whitney class
$$...
- 2,147
9
votes
1
answer
372
views
A strong form of Mostow rigidity without geometrization?
Mostow rigidity theorem says that two closed hyperbolic manifolds with isomorphic fundamental groups are isometric.
Here is my question: suppose that $M$ and $N$ are two closed 3-manifolds such that $...
- 855
7
votes
1
answer
363
views
Inverse Hodge and inverse Betti problems for Kähler manifolds
A Betti sequence is a map $\mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0}$.
A Betti sequence $b$ is realizable if there is a connected closed Kähler manifold $M$ such that $b(k)=b_k(M)$.
A Hodge diamond ...
user164740
3
votes
1
answer
257
views
Deformation equivalent Hodge structures
An HH type is the oriented homotopy type of a closed simply-connected Kähler manifold together with the Hodge structure on cohomology.
Two HH types are deformation equivalent if they are represented ...
user164740
5
votes
1
answer
394
views
Arbitrary torsion in cohomology of Kähler manifolds
Fix an integer $i\geq 3$ and a finite abelian group $G$.
Is there a connected closed Kähler manifold $M$ such that $H^i(M, \mathbb{Z})\approx \mathbb{Z}^n\oplus G$ for some integer $n\geq 0$?
user164740
3
votes
0
answers
55
views
Infinitely many deformation equivalent Hodge diamonds II
Let $S$ be a connected smooth complex-analytic space. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
...
user164740
5
votes
0
answers
130
views
Minimal sum of Betti numbers of Kähler manifold with trivial canonical bundle
Let $M$ be a closed Kähler manifold of real dimension $2n$. Suppose the canonical bundle of $M$ is holomorphically trivial.
Is it true that $\sum_{i=0}^{2n} b_i(M)=n+3\implies n=1$?
user164740
3
votes
0
answers
86
views
Exotic smooth structures on HK manifolds
An HK manifold is a closed simply-connected Kähler manifold $M$ such that $H^0(M, \Omega_M^2)=\mathbb{C}\omega$, where $\omega$ is a holomorphic 2-form on $M$ which is nowhere degenerate as a skew-...
user164740
12
votes
2
answers
767
views
Unique almost complex structure up to diffeomorphism
For which closed smooth manifolds does the action of the diffeomorphism group on the set of almost complex structures have exactly one orbit?
For example it is true for $S^2$.
user164740
3
votes
0
answers
71
views
Holomorphic homeomorphisms
Let $M$ be a connected closed smooth manifold. Consider the group $\mathrm{Homeo}(M)$ of homeomorphisms $M\to M$ endowed with the $C^0$-topology.
If $M$ has a symplectic structure some people study ...
user164740
3
votes
1
answer
124
views
Homotoping diffeomorphism to a $J$-holomorphic one
Let $M$ be a closed simply-connected smooth manifold. Assume $M$ admits at least one almost complex structure.
Is any diffeomorphism $M\to M$ homotopic as a continuous map to a $J$-holomorphic ...
user164740
6
votes
1
answer
463
views
Antiholomorphic involution with a fixed point
Let $M$ be a connected closed complex manifold. Assume it has an antiholomorphic involution. Must it have an antiholomorphic involution with a fixed point?
user164740
18
votes
0
answers
496
views
Orientation-reversing homotopy equivalence
If there is an orientation-reversing homotopy equivalence on a closed simply-connected orientable manifold is there an orientation-reversing homeomorphism?
It is not true, for instance, that if there ...
user164740
3
votes
0
answers
98
views
Non-diffeomorphic surface bundles over homeomorphic 4-manifolds
For a smooth manifold $M$ an $M$-surface is the total space of a smooth surface bundle over $M$.
Let $M_1$ and $M_2$ be two homeomorphic closed simply-connected smooth 4-manifolds. Can there be an $...
user164740
3
votes
0
answers
162
views
Exotic smooth structures on Calabi-Yau manifolds
A Calabi-Yau manifold is a simply-connected closed Kähler manifold with holomorphically trivial canonical bundle and $h^{2, 0}=0$.
If two Calabi-Yau manifolds are homeomorphic are they diffeomorphic?
user164740
13
votes
0
answers
319
views
Exotic smooth structures on Fano manifolds
If two Fano projective manifolds are homeomorphic are they diffeomorphic?
There are examples with one manifold being Fano and the other of general type (Barlow surfaces). Moreover, the number of ...
user164740
4
votes
1
answer
378
views
Chern/Hodge numbers of the conjugate complex manifold
Let $(M, J)$ be a complex projective manifold. Can $(M, -J)$ have different Chern/Hodge numbers?
user164740
5
votes
1
answer
241
views
Fano manifold becoming general type upon conjugation
Let $(M, J)$ be a Fano projective manifold. Can $(M, -J)$ be general type?
For complex curves and surfaces Kodaira dimension is diffeomorphism invariant so this cannot happen.
user164740
2
votes
1
answer
652
views
Complex manifold defined over $\mathbb{R}$
Let $M$ be a connected closed complex manifold with an antiholomorphic involution.
Must there be an atlas and a choice of a reference point in each chart such that the transition functions are ratios ...
user164740
4
votes
1
answer
447
views
Complex projective manifold with an antiholomorphic involution
Let $M$ be a complex projective manifold with an antiholomorphic involution. Can $M$ be defined by equations with real coefficients then?
user164740