All Questions
151 questions
4
votes
0
answers
362
views
Weil Kostant Integrality Result as Stated by Brylisnki
I'm reading through Brylinski's "Loop Spaces, Characteristic Classes and Geometric Quantization" and I am stuck on a piece of Theorem 2.2.15, which asserts that If $K$ is a closed complex-valued 2-...
- 1,611
4
votes
1
answer
371
views
Compatible reductions of the structure group of a principal fiber bundle
Let $P$ be a principal bundle over a manifold $M$ with structure group the Lie group $G$. Assume that $P$ admits to distinct topological reductions, say $Q_{1}$ and $Q_{2}$, where $Q_{a}$, $a=1,2$ are ...
- 2,816
3
votes
1
answer
985
views
Closed Poincaré dual, why $\int_M \omega \wedge \eta_S$ and not $\int_M \eta_S \wedge \omega $?
My book is Differential Forms in Algebraic Topology by Loring W. Tu and Raoul Bott of which An Introduction to Manifolds by Loring W. Tu is a prequel.
The characterization of the closed Poincaré dual ...
- 315
3
votes
1
answer
493
views
Four-dimensional vector bundles over $S^4$, intuition?
I know that $\pi_3(SO(4)) = \mathbb{Z} \oplus \mathbb{Z}$. We can choose an explicit identification as follows: given $(i, j) \in \mathbb{Z}$, we have a map $\phi: S^3 \to SO(4)$ which sends a unit ...
- 31
3
votes
1
answer
254
views
A category of manifolds that includes Polygonal domains
The prime motivation to introduce the category of manifolds with corners is to have a convenient theory for the analysis on simplices that is as powerful as for smooth manifolds (with boundaries).
As ...
- 5,327
3
votes
1
answer
485
views
How to compute the index of a vectorfield defined by analytic formula?
An analytic local map (or map germ) $f: (\mathbb{R}^n, 0) \to (\mathbb{R}^n, 0) $ can be considered as a vector field with zero at the origin. Assume that the origin is an isolated zero of $f$. How ...
- 194
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
3
votes
1
answer
270
views
Holonomic splitting
I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
- 35
3
votes
1
answer
458
views
Spin structures and divisibility of cohomology classes
Let me begin with some motivation. In calculating the Chern-Simons invariant of a $U(1)$ connection $A$ on a 3-manifold $M$, we can proceed by picking a bounding 4-manifold $X$ with $\partial X = M$ ...
- 631
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
127
views
Methods for constructing or checking for nontrivial classes in de Rham cohomology with local coefficients
Let $M$ be a smooth manifold (possibly with boundary), $E \to M$ a flat vector bundle, and $\mathcal{L}$ the corresponding sheaf of parallel sections.
Given a de Rham cohomology class $[\omega] \in H^...
- 6,025
3
votes
0
answers
593
views
Can additivity of the Euler characteristic be interpreted in terms of the Poincaré–Hopf theorem? [closed]
Whenever there is a long exact sequence in homology induced by a short exact sequence of chain complexes one finds that the corresponding Euler characteristics are additive. For example, if $Y \subset ...
- 31
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
2
votes
1
answer
543
views
Wilking's connectivity theorem
Wilking's connectivity theorem says, $X$ is a positively curved manifold and $Y$ is a totally geodesic submanifold of codimension $k$,then $Y$ is $n-2k+1$ connected in $X$.Then follow the theorem can ...
2
votes
1
answer
566
views
About the homotopy type of diffeomorphism groups
In this paper by Antonelli, Burghelea and Kahn (Topology, 1972), a homomorphism $L :\pi_i(\operatorname{Diff}(S^n, D_+^{n})) \rightarrow \Gamma^{n+i+1}$ was used as a tool to detect non-triviality of ...
- 111
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
2
votes
1
answer
407
views
Endomorphisms of degree d on a sphere with infinite fibers on a dense subset
Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$
disjoint n-...
- 7,609
2
votes
0
answers
208
views
Classification of bundles with fixed total space
I am aware of classification theorems for principal bundles, vector bundles, and covering spaces $\pi:E\to B$ over a fixed base space $B$. Principal and vector bundles over $B$ are classified by ...
- 501
2
votes
0
answers
137
views
Question about spin map
I'm confused with the following definition of a spin map.
A spin map is a map $f: N\to M$ between differentiable manifolds such that their second Stiefel-Whitney classes are related $\omega_2(N)=f^*\...
- 563
2
votes
0
answers
168
views
Geometric sets determined by chains (for integration and Stokes' theorem)
I have asked a similar question on mathSE more than a year ago, which received no answers, only a few comments which did not really help me. I am now re-asking this question here but reformulated ...
- 2,792
2
votes
0
answers
152
views
When are automorphisms of the cohomology ring realized by isometries?
Let $(M,g)$ be a closed smooth Riemannian manifold, and denote by $G$ a closed subgroup of its isometry group. By considering the maps $g^*$ induced by elements $g\in G$ in the (de Rham) cohomology $H^...
- 5,891
2
votes
0
answers
132
views
Do we have estimate like $\int_\gamma \alpha \le |\alpha| \cdot |\gamma|$? [closed]
Let $(X,g)$ be a compact smooth Riemannian manifold. It is known that $H^1(X, \mathbb R)\cong \mathrm{Hom} (\pi_1(X), \mathbb R)$, namely there is a natural pairing
$$
H^1(X) \times \pi_1(X) \to \...
- 2,789
2
votes
0
answers
305
views
Are homotopy equivalent manifolds with homeomorpic boundaries themselves homeomorphic?
Let $f:M \to M′$ be a homotopy equivalence of topological manifolds with boundary such that $dim(M)=dim(M′)$ and $f:\partial M \to \partial M′$ is a homeomorphism. Does this imply the existence of a ...
- 79
2
votes
0
answers
188
views
How many linear independent vector fields can be constructed on a general manifold with $\chi(M)=0$?
We have known how many linear independent vector fields can be constructed on $S^n$:https://en.wikipedia.org/wiki/Vector_fields_on_spheres
So how many linear independent vector fields can be ...
- 977
2
votes
0
answers
385
views
Intuition behind the following theorem of Reeb?
What is the intuition behind the following theorem of Reeb?
If a compact manifold admits a function with only two critical points which are non degenerate, it is homeomorphic to the sphere.
- 29
1
vote
1
answer
206
views
A p-form taking discrete values on p-chains must be 0.
I want to show that if $w$ is a $p$-form such that its induced cochain on $p$-chains:
$w(\gamma)= \int_{\gamma} w \in S$
takes values in a discrete set $S \subset \mathbb{R}$ then $w$ must be zero.
...
- 1,611
1
vote
1
answer
924
views
Normal tubular neighborhood theorem for semi(or pseudo)-riemannian manifolds
Suppose you have a manifold $M$ and a closed sub-manifold $A$, and let $g$ be a semi-riemannian metric,ie, $g_x$ defines a quadratic form on $T_xM$ such that $g_x(v,v)\ge0$, but $g_x(v,v)=0$ not ...
1
vote
1
answer
396
views
Orientability of Surfaces and the Fundamental Group [closed]
Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given ...
- 11
1
vote
0
answers
153
views
Poincaré-Hopf Theorem for domains with a point of vanishing curvature
Consider $\Omega \subset \mathbb{R}^2$ a convex planar domain having positive curvature on the boundary except for a point $p \in \partial \Omega$ where the curvature vanishes.
I would like to know ...
- 111
1
vote
0
answers
151
views
Witten's QFT Jones polynomial work on Atiyah Patodi Singer theorem and $\hat A$ genus over Chern character
In Witten's 1989 QFT and Jones polynomial paper,
he wrote in eq.2.22 that
Atiyah Patodi Singer theorem says that the combination:
$$
\frac{1}{2} \eta_{grav} + \frac{1}{12}\frac{I(g)}{2 \pi}
$$
is a ...
- 447
1
vote
0
answers
139
views
Cartesian product of spin manifolds [closed]
Is it true that the Cartesian product of two spin manifolds is spin?
1
vote
0
answers
246
views
Relation between the pushback closed form of sphere bundle and the pullback closed form of ball bundle
Let $B$ be a closed oriented $n$-manifold, and $\pi_N:N\to B$ be an oriented $m$-dim ball bundle, i.e. each fiber is an oriented $m$-dim ball(disk) $D^m$. We have a sphere bundle $\pi_\partial:\...
- 1,915
1
vote
0
answers
32
views
Results on compact slices in a regular foliation
Let $(M,\mathcal{F}$) be a smooth and regular foliation (not necessarily of comdimension 1). I am wondering if there are known (partial) results on the existence of compact, connected submanifolds $F\...
- 133
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
1
vote
0
answers
194
views
Existence of Morse function on suspension
Let $X$ be a smooth simply connected compact manifold of dimension $n$ with boundary. Let $Y$ be a smooth compact manifold of dimension $n+1$ without boundary such that $H_{i+1}(Y)=H_{i}(X)$(reduced ...
- 61
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
1
vote
0
answers
116
views
Defining the cospecialization in topology
Below is an excerpt from part V of Deligne's Étale cohomology - starting points.
Let $X$ be a complex analytic variety and $f:X\to D$ a morphism from $X$ to the disk. We denote by $[0,t]$ the ...
- 10.5k
1
vote
0
answers
205
views
Analog of Gauss-Bonnet formula for principal bundles over manifolds with boundary
The Gauss-Bonnet formula gives a topological invariant as an integral over a local density on the given manifold. In particular, when there is a boundary, GB formula has to be supplemented by a ...
- 21
1
vote
0
answers
320
views
Does the closure of a ``nice'' smooth submanifold define a homology class?
Let $M$ be a smooth compact, oriented manifold. Let
$X$ be a submanifold which is of the following type
$$X := \{ p \in M: \psi(p) =0, ~~\varphi(p) \neq 0 \} $$
where
$$ \psi: M \rightarrow V, \...
- 3,245
0
votes
1
answer
201
views
Ambient isotopy of the diagonal submanifold in product space
Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
$...
- 73
0
votes
1
answer
364
views
Proper actions and diffeomorphism groups
Since the diffeomorphism group is not locally compact; is it true that there is no proper action of an infinite-dimensional diffeomorphism group on a finite-dimensional smooth manifold?
Edit: The ...
- 111
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
0
votes
1
answer
254
views
$SU(k)/SO(k)$ as a manifold, for each positive integer $k$ [closed]
The quotient space $SU(k)/SO(k)$ is also a homogeneous space constructed out of the Lie groups (special unitary $SU(k)$ and special orthogonal $SO(k)$).
Because the $SO(k)$ may not be a normal ...
- 317
0
votes
1
answer
155
views
Vector bundles over a homotopy-equivalent fibration
I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here.
Let $\pi:N\rightarrow M$ be a smooth ...
- 2,792
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?
$$X_0\...
- 563
0
votes
1
answer
201
views
factorization morphism between projectives spaces
Please help me with this doubt:
Let $f:\mathbb{P}^1 \rightarrow \mathbb{P}^2$ be a non-constant morphism. Is there any factorization of $f$ as $$\mathbb{P}^{1} \overset{h}{\rightarrow}\mathbb{P}^{2}\...
- 25
0
votes
0
answers
85
views
Existence of covering space with trivial pullback map on $H^1$
I have seen somewhere the following claim (but can't remember where): let $M$ be a connected orientable closed smooth manifold with $b_1(M)=1$, then there exists a connected covering space $p:\tilde{M}...
0
votes
0
answers
58
views
Role of basins of attraction in the Morse decomposition
Let $M$ be a differentiable manifold and $F \in \mathcal{X}(M)$. We define a DS by
$$\dot{x}=F(x(t))$$
An ordered collection $\mathcal{M}=\left\{M_{1}, \ldots, M_{l}\right\}$ of compact subsets of ...
- 247
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
-2
votes
1
answer
189
views
Topologies in the vicinity of Euclidean space
Given a smooth function $f:\mathbf R^n\to \mathbf R^m$ with $0$ as a regular value, I define the $(n-m)$ dimensional smooth manifold $M_f:=f^{-1}(0)$.
Let $f_0(x_1,...,x_n):=(x_1,...,x_m)$; $M_{f_0}$ ...
- 521