All Questions
149 questions
9
votes
2
answers
753
views
Deformation equivalent vs diffeomorphic to projective manifold
Let $M$ be a closed complex manifold that is not deformation equivalent to a complex projective manifold.
Can $M$ be orientedly diffeomorphic to a complex projective manifold? What if $M$ is moreover ...
user164740
3
votes
0
answers
109
views
Kähler manifolds deformation equivalent to projective manifolds
Let $M$ be a closed non-projective Kähler manifold. There are three possibilities
there is a proper holomorphic submersion $f:X\to \Delta$ with $f^{-1}(0)\cong M$ such that the projective fibers ...
user164740
2
votes
1
answer
128
views
Infinitely many deformation equivalent Hodge diamonds
Let $S$ be a connected open complex manifold. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
An ...
user164740
10
votes
2
answers
499
views
Symplectic structure on the square of a 3-manifold
Let $M$ be a connected closed orientable 3-manifold. Assume $M$ is not the direct product of a surface and the circle.
Can there be a symplectic or Kähler manifold homeomorphic to $M\times M$? I think ...
user164740
7
votes
1
answer
634
views
Hodge diamonds of complex threefolds
There is no closed complex curve or surface with $h^{1, 0}-h^{0, 1}=1$.
Now consider threefolds. Can this condition be satisfied?
Is Serre duality in fact the only restriction on the Hodge diamond?
user164740
11
votes
1
answer
379
views
Smooth structure on direct product
Let $M$ be the $E_8$ manifold. Is there a closed manifold $N$ such that $M\times N$ is smoothable? What is the smallest possible dimension of $N$?
user164740
10
votes
1
answer
362
views
Topological factors of complex projective manifolds
Let $M$ be a closed orientable smooth 4-manifold. Assume $\pi_1(M)=\{0\}$ and $b_2(M)>0$.
Let $S$ be a closed orientable surface. Denote $P=M\times S$.
Can it so happen that there is no complex ...
user164740
3
votes
0
answers
144
views
All Kähler threefolds embed into a common complex manifold
Is there a closed complex manifold into which all closed complex threefolds admitting a Kähler structure embed?
user164740
4
votes
0
answers
138
views
Topology of Brody hyperbolic manifolds
Let $M$ be a Brody hyperbolic complex projective manifold with $\pi_1(M)=\{0\}$. Can $M$ be homeomorphic to $P\times S^2$ where $P$ is a manifold?
user164740
7
votes
1
answer
238
views
Diffeomorphisms pushing forward vector field to any non-zero multiple
Is there a closed smooth manifold $M$ such that for each real $x\neq 0$ there is a nowhere vanishing vector field $v$ on $M$ and a diffeomorphism $\phi:M\to M$ such that $\phi_*v=xv$?
user164740
2
votes
1
answer
263
views
Smooth covers rescaling the symplectic form
Let $(M, \omega)$ be a connected closed symplectic manifold of dimension $2n$.
Assume there exist smooth covering maps $\phi_k:M\to M$ such that $\phi_k^* \omega=\sqrt[n]{k}\omega$ for all $k\geq 1$. ...
user164740
1
vote
0
answers
137
views
Covers of a 4-manifold pull back a cohomology class to any algebraic multiple
Fix an algebraic integer $x\neq 0$. Is there a closed smooth 4-manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$?
Is ...
user164740
4
votes
1
answer
222
views
Smooth covers pulling back a cohomology class to any algebraic multiple
Fix an algebraic integer $x\neq 0$. Does there exist a closed smooth manifold $M$ with a class $\rho\in H^{1}_{\mathrm {dR} }(M)$ and a smooth covering map $\phi:M\to M$ such that $\phi^*\rho=x\rho$?
user164740
5
votes
1
answer
184
views
Fundamental groups of primitive non-algebraic compact Kähler manifolds
Call a compact topological manifold $M$ primitive if there is no Serre fibration $M\to B$ where $B$ is a CW complex of dimension $0<d<\mathrm{dim}(M)$.
Given a Kähler group $G$ does there exist ...
user164740
8
votes
1
answer
452
views
Non-algebraic Kähler threefolds with abelian $\pi_1$ of arbitrarily large rank
Do there exist non-algebraic Kähler threefolds with abelian $\pi_1$ of arbitrarily large rank?
user164740
4
votes
2
answers
405
views
Conformal covers of all degrees
Let $M$ be a connected closed conformal oriented manifold.
Assume there exist conformal covering maps $\phi_k:M\to M$ of all degrees $k\geq 1$. Is $M\cong S^1$ then?
Can we at least rule out $\mathrm{...
user164740
32
votes
3
answers
1k
views
Complex projective manifolds are homeomorphic if homotopy equivalent
If two complex projective manifolds are homotopy equivalent are they homeomorphic?
user164740
4
votes
0
answers
273
views
Instanton numbers for diverse gauge bundles on diverse manifolds --- their relations to characteristic classes
It is standard (?) that the $SU(N)$ gauge theory has the instanton number $n$ quantized as $n \in \mathbb{Z}$
$$
n = { 1 \over 8\pi^2} \int_{\mathcal{M}_{4}} \text{tr} \left(F \wedge F\right) = {1 \...
- 10.5k
3
votes
0
answers
186
views
Cobordism theory of some weird space
Let $G=SU(3)$ and $N=SO(3)$, then $G/N= SU(3)/SO(3)$ = a 5-dimensional Wu manifold $W$.
The $W$ is a homogeneous space (also a quotient space), but not a group.
Previously, I am aware of the ...
- 10.5k
5
votes
1
answer
389
views
Lengths of closed geodesics on a flat vs hyperbolic punctured torus
Let $T$ be a torus (oriented closed surface of genus 1), $p\in T$, and $T^* := T - \{p\}$.
Let $\mu$ denote a flat structure on $T$. This can be obtained for example by choosing a uniformization $p_f:...
- 10.7k
5
votes
1
answer
248
views
Multisignature and homeomorphism type
In classical surgery theory, there is a map
$$L_{n+1}(\pi_1M)\to S(M^n)$$
Element in $L_{n+1}(\pi_1M)$ is realized as surgery obstruction of a surgery problem to $M\times I$ with one boundary piece ...
- 101
4
votes
1
answer
198
views
Space of non-vanshing sections path-connected?
Let $M$ be a path connected smooth manifold and $E$ be a vectorbundle over $M$ of rank at least two. My question is: Under which conditions is the space of global non-vanishing sections path connected?...
- 271
7
votes
0
answers
226
views
The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons
In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical
group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with ...
- 10.5k
5
votes
1
answer
503
views
Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure
The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) ...
- 10.5k
16
votes
2
answers
605
views
What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?
The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. ...
- 63.9k
6
votes
1
answer
679
views
Generalized projective spaces, spheres, and exotic spheres [closed]
I like to explore and ask for proper references for the relations between generalized projective spaces, spheres, and exotic spheres:
The real projective space
$\mathbb{RP}^1 \simeq S^1,$
is ...
- 10.5k
7
votes
1
answer
223
views
Five-dimensional manifolds fibering over a fixed hyperbolic surface
I am aware of the classical work by Smale and Barden computing the diffeomorphism type of smooth simply connected 5-manifolds in D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965),...
- 2,630
5
votes
1
answer
266
views
$S^1$-quotient of the space of unbased contractible loops of a finite dimensional $K(\pi,1)$
Let $X$ be a finite dimensional $K(\pi,1)$ manifold. Recall that the space of unbased null-homotopic loops $L_0(X)$ is the space of contractible continuous maps $S^{1}\to X$. There is an obvious $S^1$-...
- 14.3k
5
votes
1
answer
388
views
The space of contractible loops of a finite dimensional $K(\pi,1)$
Let $X$ be a finite dimensional $K(\pi,1)$ manifold. Is it true that the space contractible loops of this manifold can be contracted to the space of constant loops on $X$? What if $X$ is a finite ...
- 14.3k
10
votes
1
answer
1k
views
Classification of oriented vector bundles of rank 5 over closed oriented 5-manifolds
I am looking for a complete classification in terms of characteristic classes and "computable" (preferably geometric) invariants. There is this work where the authors classify oriented vector bundles ...
- 2,830
2
votes
1
answer
140
views
Could an inverse of (weak) Morse inequality exists in some special case?
Could an inverse Morse inequality hold in some sense? More precisely I wish the following result to be true:
Problem
$M$ is a smooth simply connected compact manifold, $dim(M)=n$, $f$ is a morse ...
- 697
16
votes
3
answers
940
views
Relationships between homology maps of cobordant manifolds
Let $W$ be a cobordism between $n$-manifolds $M$ and $N$, and $f\colon W \mapsto X$ be a map to some manifold $X$.
Does anybody know of any nice examples of general relationships between the images ...
- 161
3
votes
0
answers
75
views
Two questions regarding flat fibre bundles and the corresponding group action on the fibre
Let $F$, $B$ be smooth, closed manifolds and $\phi:\pi_1(B) \rightarrow Aut(F)$ a smooth group action of the fundamental group of $B$ on $F$.
Consider the flat fibre bundle $E_\phi := \widetilde{B} \...
- 404
11
votes
0
answers
650
views
Triangulation of manifolds with corners
Let's begin with some definitions:
A (smooth) manifold with corners is a Hausdroff (and second countable if you want) space that can be covered by open sets homeomorphic to $\mathbb R^{n-m} \times \...
- 909
10
votes
2
answers
730
views
Representability of the sum of homology classes
This is probably a very simple question, but I have not found it addressed in the references that I know. Let $M$ be a closed and connected orientable $d$-dimensional manifold ($d\leq 8$) and let $[\...
- 2,816
16
votes
3
answers
1k
views
SO(3) action on (simply connected) 6 manifold with discrete fixed point
If a 6-dimensional orientable smooth manifold $M$ admits a smooth effective $SO(3)$ action with discrete fixed point set, what can we say about the topology of $M$? What if we assume that in addition ...
- 591
3
votes
1
answer
115
views
Decomposing global isotopies into local ones
I'm looking for a reference for the following basic-looking statement:
Let $X$ be a smooth manifold covered by open sets $U_1$ and $U_2$. Let $f:X \rightarrow X$ be a map isotopic to identity via an ...
- 3,772
0
votes
1
answer
395
views
Topology of manifolds and transition functions
let me start by describing some examples which may well demonstrate the motivation:
A manifold is obtained by glueing together Euclidean spaces, and there is a transition function on the overlap of ...
- 1
6
votes
1
answer
456
views
Restrictions on $\pi_1(X)$ of geometric origin (Kähler groups as example)
There's and old and extensively studied question about characterisation of fundamental groups of smooth compact Kähler manifolds. Restrictions imposed by Kählerness are somewhat fragile, and if we ...
- 4,600
6
votes
2
answers
817
views
Can a Morse function define a unique structure on a closed manifold?
I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have ...
- 3,751
4
votes
2
answers
619
views
Is it true that all sphere bundles are some double of disk bundle?
Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...
6
votes
1
answer
417
views
"structure group" for fibration
Regarding "fibration" as a homotopy analogue of "fiber bundle",I want to see parallel notions of "structure group" and "fiber change" in "fibration".
Does it make sense to talk about "structure group"...
- 71
3
votes
1
answer
351
views
Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n
For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two, $\pi_1$-surjective, ...
- 113
6
votes
0
answers
232
views
Does $\#_n S^2×S^1$ really admit a map of non-zero degree from $B×S^1$?
In Proposition 4 on page 6 of this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ ...
- 61
18
votes
1
answer
1k
views
Is the restriction map for embeddings of manifolds with boundary a fibration?
Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.
Richard ...
- 2,974
9
votes
1
answer
384
views
embedding of quaternionic projective spaces
Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding
$$
\mathbb{H}P^2\longrightarrow \mathbb{R}^N?
$$
Are there any ...
- 1,990
2
votes
1
answer
1k
views
embeddings of product of spheres in Euclidean spaces [closed]
I notice that $T^2=S^1\times S^1$ can be embedded in $\mathbb{R}^3$ as a hypersurface (submnaifolds of codimension 1).
In general,
(1). could the product of spheres $S^{m_1}\times\cdots\times S^{...
- 2,223
0
votes
0
answers
134
views
when is "fibering" preserved under homotopy equivalence
Suppose I have an oriented $F$ bundle over $B$ with total space $E$ (all of the three are closed manifolds) and i have a closed manifold $E'$ which is homotopy equivalent to $E$.Is there any condition ...
- 259
6
votes
1
answer
260
views
Fundamental class in $KO[1/2]$
Let $M^m$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:
Rationally, $\Delta_M$ is ...
- 61
18
votes
2
answers
1k
views
formula for Eta invariant
Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
$$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet ...
- 259