12
votes
0answers
152 views

What do tangles teach us about braids?

A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times ...
0
votes
0answers
34 views

General information on sets of pairwise transverse submanifolds

I am looking for general information on the following definition: Given a manifold $M$, let $T_k(M)$ be the set containing all sets of pairwise transversal submanifolds on $M$ of dimension $k$. So ...
6
votes
1answer
191 views

cell decomposition of real homogeneous hypersurfaces

Let $f(x_1,\ldots,x_n)\in\mathbf{R}[x_1,\ldots,x_n]$ be a homogeneous polynomial and consider the real hypersurface $H=\{(x_1,\ldots,x_n)\in\mathbf{R}^n:f(x_1,\ldots,x_n)=1\}$. Assume to simplify ...
3
votes
2answers
141 views

Circle Bundles of surfaces

Let S be a surface with a metric of constant curvature and finite area. Is there a classification of the circle bundles over S?
2
votes
2answers
203 views

manifolds with unusual rational cohomology rings

I'm looking for examples of 3-manifolds with unusual rational cohomology rings. I'm curious about what the cup product structure can actually look like, and I'd like some examples to play with. Does ...
1
vote
0answers
158 views

Does there exist any subspace of R^n, homeomorhic to a manifold but not a C^0 submanifold of R^n?

At first I thought that if a subspace of $\mathbb{R}^n$ is homeomorphic to a manifold, then it is a $C^0$ submanifold of $\mathbb{R}^n$. But I found an asterisked exercise in the book Differential ...
8
votes
2answers
231 views

The boundary of a domain whose interior is diffeomorphic to the ball

We know that there are compact manifolds with diffeomorphic interiors but their boundaries are not homeomorphic (see the question Manifolds with homeomorphic interiors). My question is about a very ...
-1
votes
2answers
166 views

Let be $f \in Diff(M)$. What we can say about the subgroup $span{f}< Diff(M)$? What are implications in the structure of $f$ and $M$? [closed]

Let be $f \in Diff(M)$. When is finite the subgroup $span\{f\}< Diff(M)$? What are implications in the structure of $f$ and $M$?
3
votes
1answer
203 views

Is any smooth homeomorphism isotopic to a smooth embedding?

Let $f:D^m\to \mathbb{R}^m$ be a smooth map ($D^m$ is the unit ball). We call $f$ embedding if it is a homeomorphism on the image and the derivative $D_xf$ is nonsingular at each point $x\in D^m$ ...
4
votes
0answers
91 views

are smooth homotopic open embedding of Hilbert manifolds smoothly isotopic?

A theorem of Chapman (Chapman, T. A. Homotopic homeomorphisms of infinite-dimensional manifolds. Compositio Math. 27 (1973), 135–140) states that every two open embeddings $f,g:M\to N$ of Hilbert ...
7
votes
0answers
174 views

Reference Request: Topological h-cobordism theorem in higher dimensions

I think this question on math.stackexchange is more appropriate on mathoverflow. Correct me, if you don't think so. The h-cobordism theorem is true in the topological and in the smooth category in ...
4
votes
1answer
171 views

Codimension zero embeddings and diffeomorphism groups

Let $V$ be a smooth manifold obtained by attaching the ``open collar'' $[0,1)\times \partial N$ to a compact smooth manifold $N$ along the boundary. Let $\mathrm{Emb}(N, V)$ be the space of embeddings ...
23
votes
8answers
2k views

Are there some other notions of “curvature” which measure how space curves?

I am learning differential geometry and have a few questions on curvature. -- Background: Gauss invented "Gauss curvature" to measure how surface curves. Riemann gives an ingenious generalization ...
0
votes
3answers
325 views

Good books on Geometric Theory of Dynamical Systems

I am looking for a good book on Geometric Theory of Dynamical Systems . I found Geometric Theory of Dynamical Systems by Jr. Palis myself,but it's very old, anyway i would like to find a pure ...
6
votes
1answer
331 views

consequence of Novikov conjecture

Novikov conjeture is a famous open problem in Geometric topology.It predicts that higher signature is oriented-homotopy invariant. http://en.wikipedia.org/wiki/Novikov_conjecture I am a student ...
3
votes
0answers
204 views

Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...
0
votes
1answer
85 views

Connecting two hypersurfaces in R^{n+1} by embedded curves

Let $M^n$ be a smooth closed embedded hupersurface in $\mathbb R^{n+1}$. Denote by $D$ the bounded connected component of $\mathbb R^{n+1}\backslash M$. We assume that $\mathbb R^{n+1}\backslash D$ is ...
0
votes
0answers
167 views

Fractional degree of a map?

Is there some natural notion of a fractional degree of a map? The degree of a map is a generalization of the winding number, and fractional winding numbers appear in the (mathematical physics) ...
9
votes
0answers
183 views

Embedding of products of projective spaces into a projective space

Does anybody know estimates on the minimal dimension $k$ for which the product $P^n \times P^m$ can be smoothly embedded into $P^k$? I am interested in projective spaces over $\mathbb RR$ and $\mathbb ...
2
votes
0answers
161 views

Stratification of a smooth map

So, this is an exercise. But from math.stackexchange I have been suggested to post this question here. To find the Thom-Boardman stratification of the smooth map ...
5
votes
2answers
288 views

Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization?

The question is in the title. The form of the condition looks like the Bohr-Sommerfeld quantization formula of angular momentum, is there a link between the two formulas?
4
votes
0answers
199 views

Is there an “exponential law” for differentiable maps between smooth manifolds?

Although it seems like a textbook question, I was not able to find a textbook or even a research article answering the following question: Let $M$, $N$ and $P$ be finite-dimensional smooth manifolds ...
3
votes
1answer
144 views

Lipschitz Approximation to a PW Smooth Map

Suppose I have a triangulated smooth manifold, $\tau : |K| \rightarrow M$ (so that $\tau | _{\sigma}$ is smooth for each $\sigma \in K$), and a piecewise smooth map, $f: M \rightarrow \mathbb{R}^n$. ...
2
votes
2answers
130 views

intersection of Whitney stratifications

Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is ...
12
votes
2answers
811 views

Converse of Poincaré-Hopf theorem

Let $M$ be a connected, compact, oriented manifold of dimension $n<7$. If any two maps $M \to M$ having equal degrees are homotopic, must $M$ be diffeomorphic to the $n$-sphere?
15
votes
0answers
403 views

Do there exist exotic 4-tori?

More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds? Related: if such a ...
9
votes
0answers
286 views

3 manifolds with diffeomorphic unit tangent bundles

What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?
5
votes
0answers
230 views

Quotient of 3-sphere by binary octahedral group?

Consider the Lie group $Spin(3)$, which can be thought of geometrically as the 3-sphere (e.g., it can be represented by the collection of unit quaternions). The quotient $Spin(3)/\pm I$ yields the ...
8
votes
1answer
341 views

Construction of exotic spheres that do not bound parallelizable manifolds

There are at least two ways to construct homotopy spheres that bound parallelizable manifolds, namely Milnor's plumbing construction and Brieskorn's method of singularities, and each of these methods ...
5
votes
0answers
211 views

Existence of particular embeddings in euclidean spaces for non compact manifolds

Let $M$ be a $n$-dimensional smooth non-compact manifold such that the singular cohomology groups $H^{k}(M,\mathbb{Z})$ are finitely generated for $k\geq 0$. Can we find a sufficiently big integer ...
12
votes
1answer
570 views

Strong Whitney embedding theorem for non-compact manifolds

$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds. The strong ...
1
vote
0answers
130 views

Uniqueness of the Smooth Structure on a Handle Attachment [closed]

I posted this question on math stack exchange and didn't receive an answer. If it is too elementary for this forum I will be happy to delete it. Let $M^m$ be a smooth manifold with boundary. We may ...
7
votes
1answer
464 views

Cancellation law for $M^n\times \mathbb R= N^n\times \mathbb R$.

Assume $M^n$ and $N^n$ are null bordant, i.e. each can be realized as boundary of an $n+1$ dimensional manifold. Suppose $M^n \times \mathbb R$ is homeomorphic to $N^n\times \mathbb R$. Is there any ...
10
votes
1answer
278 views

What is the status of the PL-pseudoisotopy stability theorem?

Suppose that $M$ is a compact PL-manifold (possibly with boundary) and let $C^{PL}(M)$ denote the (simplicial) group of PL isomorphisms of $M \times I$ relative to $M \times \{0\} \cup \partial M ...
17
votes
3answers
1k views

4D TQFT from a modular tensor category

I know the construction of a 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
6
votes
1answer
298 views

Relation between TQFT and Wilson lines, boundary conditions, surface defects etc

I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered ...
0
votes
1answer
288 views

Diffeomorphism coming from the s-cobordism theorem

Let $f:M_0\rightarrow M_1$ be a diffeomorphism between two compact $n$-dimensional manifolds. Let $W^{n+1}$ be an $h$-cobordism between $M_0$ and $M_1$. Assume that the cobordism has no torsion and ...
2
votes
4answers
817 views

Line bundle on $S^2$

How do you prove that a line bundle (vector bundle of rank 1) on $S^2$ is isomorphic to the trivial line bundle? Can you give a reference? Thanks.
4
votes
2answers
486 views

Intersection forms of 4-manifolds with boundary

Let $X$ be any simply connected smooth 4-manifold with a fixed Euler characteristic $e$, signature $\sigma$ and boundary $Y$. Assume that the determinant of the intersection form $Q_{X}$ is equal to a ...
0
votes
1answer
194 views

Diffeomorphisms of a surface in terms of generators.

I am interesting in a presentation of a diffeomorphisms in terms of generators. Is it possible to obtain such presentation in some cases, depending on a genus of a surface or a type of diffeomorphism ...
12
votes
1answer
1k views

Kervaire invariant: Why dimension 126 especially difficult?

Is there any resource that might help non-experts gains some understanding of why the Kervaire invariant problem remains open now only in dimension $126$? ($126 =2^7-2=2^{j+1}-2$; whether ...
4
votes
0answers
421 views

Topological version of two results in smooth Morse theory

Morse theory is generally presented in the DIFF category. However, there is a version of Morse theory in TOP (see the post Morse theory in TOP and PL categories? for references). It is well known ...
12
votes
1answer
447 views

Handlebody decomposition of an open 4-manifold

Let $M$ be the fake $CP^2$ (namely the closed topological 4-manifold which is homotopy equivalent but not homeomorphic to the complex projective plan). It is well-known that $M$ admits no smooth ...
20
votes
2answers
622 views

Manifolds with homeomorphic interiors

Suppose that two compact topological manifolds with boundary have homeomorphic interiors. Can we conclude that the two manifolds are homeomorphic? What happens in the smooth category?
4
votes
1answer
292 views

How to compute the Monopole Floer Homology for Surface $\times S^1$ ?

We know that Monopole Floer homology of a 3-manifold $M$ depends on a spin-c structure. My question is that if $M$ is $F\times S^1$ ($F$ is a surface of genus larger than 1) then how can we compute ...
9
votes
2answers
991 views

When does the tangent bundle of a manifold admit a flat connection?

Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$? Edit: Formerly, I asked about a flat connection on the frame ...
5
votes
3answers
876 views

Where can I find a full proof of the Chern-Gauss-Bonnet theorem ?

Hello, I am looking for a proof for the Chern-Gauss-Bonnet theorem. All I have found so far that I find satisfactory is a proof that the euler class defined via Chern-Weil theory is equal to the ...
5
votes
1answer
249 views

Characteristic Classes of a Fibered Sum

I will phrase this question in terms of attaching smooth manifolds along a submanifold, though it is certainly more general. Let $M_1$ and $M_2$ be smooth $n$-manifolds (maybe closed, for ...
17
votes
2answers
990 views

The difference between a handle decomposition and a CW decomposition

Let $M$ be a compact finite-dimensional smooth manifold. I have a question about the relationship between the statements that a Morse function induces a handle decomposition for $M$, and that it ...
4
votes
1answer
308 views

Existence of integrals of 1-forms up to multiples

Let $\Omega\subset \mathbb{R}^n$ an open contractible set (we can assume $n=2$ for a start) and $\omega$ be a 1-form on $\Omega$ which is nowhere zero. Then $\omega=df$ for a function $f$ if and only ...