**9**

votes

**1**answer

243 views

### Reference request: Topology on the space of smooth compact submanifolds

In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ ...

**3**

votes

**0**answers

54 views

### Connected sum of chiral manifolds

Let $M,N$ be two closed, smooth, orientable manifolds of the same dimension and assume that these manifolds are chiral, i.e. they do not admit an orientation reversing automorphism. Then there are two ...

**3**

votes

**0**answers

81 views

### Multiplicativity of combinatorial l classes

For closed smooth manifolds $M$ and $N$, the Hirzebruch $L$ class is multiplicative, i.e. $L(M\times N)=L(M)L(N)$. Is this property still true if $M$ and $N$ are assumed to be closed topological ...

**2**

votes

**0**answers

106 views

### Invariant generalized sections of dual vector bundles

Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My ...

**0**

votes

**0**answers

59 views

### What does deg=0 imply for the gauss map?

I am facing the following problem. I have a Riemannian manifold $(M,g)$ with gauss curvature zero, an isometric immersion $v:M\rightarrow \mathbb{R}^3$ that is $C^{1,\alpha}$ and I consider the Gauss ...

**4**

votes

**1**answer

183 views

### Are “Unions” of small exotic $\mathbb{R}^4$'s small?

Suppose $M$ is a smooth 4-manifold, and $U,V \subset M$ are exotic $\mathbb{R}^4$'s, i.e. homeomorphic to standard $\mathbb{R} ^4$.
Further more suppose $U$ an $V$ intersect nicely sucht that $U \cup ...

**2**

votes

**1**answer

101 views

### Intersection of two real polynomial surfaces

Consider two real polynomials in three variables, defined on the 3-sphere, $S^3$. Is there some Bezout-type theorem, relating the intersection of two closed surfaces defined by these polynomials and ...

**0**

votes

**0**answers

39 views

### Disturbing regular level submanifold of a smooth function

Let $a$ be a regular value of a smooth function on a closed manifold and $\{f=a\}$ a corresponding level submanifold. It is known that any such function can be approximated by a Morse function $g$. ...

**1**

vote

**1**answer

110 views

### Generalized spin connection and dreibein in higher spin gravity

I am studying higher spin gravity and I would like to know the mathematical and physical meaning of generalized spin connection and generalized dreibein that appear in this theory.
It is well known ...

**1**

vote

**2**answers

136 views

### Local structure of the quotient of a Lie group action

Suppose $M$ is a smooth manifold and a compact Lie group $G$ acts freely on $M$, then it is well known that $M/G$ has a manifold structure.
Are there any results for the general case? (a) If the ...

**1**

vote

**0**answers

66 views

### Determine the shape of curve which has the minimum number of inflection points

Let $A$ be a set of generic $C^2$ closed curves in $2$-dimentional Euclidean space, and an equivalence relation between $2$ curves $a$ and $b$ in $A$ is defind that there exist diffeomorpfic map $h ...

**0**

votes

**1**answer

73 views

### Set of critical values is compact [closed]

Let $M\subseteq \mathbb{R}^n$ be a compact manifold with $\partial M=\emptyset$ and let $f: M\rightarrow S^p$ be a smooth map. I want to unerstand the proof of the following lemma which occurs in ...

**4**

votes

**1**answer

328 views

### Cap product à la Poincaré

Recently, it became apparent to me that I was not the only one who always first thought in terms of cap product before actually computing a cup product. There is no denying this is evil, but I found ...

**4**

votes

**1**answer

304 views

### Is the space of real conics with a singular point an orientable manifold?

Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set
of such a polynomial gives a real curve ...

**0**

votes

**1**answer

110 views

### Surjectivity of “nice maps” from local properties

What tools are available from real algebraic geometry, analysis and
topology to check surjectivity of a map $f:M_{1}\rightarrow\mathbb{R}^{d}$
from local properties and maybe function values?
...

**3**

votes

**1**answer

375 views

### What is wrong with the “naive” proof of the Hauptvermutung?

The Hauptvermutung is the statement that any two PL structures on a topological space have a common refinement. It is false in general, but (I think) true for some low dimensional manifolds.
The ...

**2**

votes

**1**answer

156 views

### Axiomatization of Degree Theory

I am reading Ruiz's Mapping Degree Theory, and find an axiomatization of degree theory of $\mathbb R^n$ in P38. It says that there exists a unique map $d(f,D,y)\in\mathbb Z$ satisfies
Normality
...

**2**

votes

**0**answers

143 views

### Example of symplectic 4-manifolds with no Lefschetz fibration structure?

I just read about Donaldson's result on existence of Lefschetz pencil structure on symplectic manifolds (Donaldson 1999). However, one has to blow up the base locus to get a Lefschetz fibration ...

**8**

votes

**2**answers

428 views

### Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated?

Are there some tables or handbooks of homology and homotopy groups of every manifold which has been calculated? Or are there some tables or handbooks which list some common calculated results of ...

**19**

votes

**0**answers

256 views

### Can one properly embed a differential manifold into numerical space of double dimension? [duplicate]

If $X$ is a $ C^\infty$ differential manifold of dimension $n$, then there exists an embedding $f:X\to \mathbb R^{2n+1}$.
This is a not too difficult theorem due to Whitney, proved in many textbooks.
...

**-1**

votes

**1**answer

195 views

### Are compact complete geodesics closed? [closed]

note: I find this question In stackexchange math, I would be interest to know how I could be answer this kind of question,I pasted it here as I see it appropriate For MO.
check this link: ...

**1**

vote

**1**answer

148 views

### Normal Variation on Manifolds

Let $M$ be a smooth surface and let $x,y \in M$. Let $d_{M}(\cdot, \cdot)$ be the geodesic distance metric on $M$, that is the length of the shortest geodesic curve on $M$. Let $\kappa$ be the maximum ...

**4**

votes

**0**answers

245 views

### If 2-manifolds are homeomorphic and smooth, are they diffeomorphic? [closed]

Perhaps this question has already been asked on Mathoverflow. I mean this question in a global sense. A friend mentioned it to me today, and I started thinking about it. I'm not sure how to prove it. ...

**4**

votes

**1**answer

58 views

### Existence of an equivariant Morse function

Let $G$ be a (finite) group and $M$ a $G$ manifold. Now I have a smooth real valued function $f: M\rightarrow R$ with $f(x)=f(g(x)),\, \forall g\in G$. Now in general $f$ will maybe not be a Morse ...

**1**

vote

**1**answer

130 views

### Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite area such that

Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite areas such that for each simply finite area there exist a diffeomorphic map that ...

**5**

votes

**0**answers

197 views

### Gompf's invariant of $2$-plane fields

I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...

**2**

votes

**1**answer

63 views

### Constructivity of zeros demanded by topological degree

Let $f : S^{n - 1} \to S^{n - 1}$ be a smooth map from the unit vectors of $\mathbb{R}^n$ to themselves. If $f$ has nonzero degree, then we know that any smooth map $g : D^n \to \mathbb{R}^n$ ...

**0**

votes

**0**answers

92 views

### Degree of Map between Pseudomanifold

There are two different ways to define a degree of map.
Let $M$, $N$ be smooth, and $M$ is compact, $N$ is connected. If $f\in C(M,N)$, we define $\deg f$ by smooth approximation. [Milnor/Topology ...

**3**

votes

**0**answers

99 views

### When does a leaf space admit a (non-Hausdorff) manifold structure?

If $f:M \to N$ is a submersion with connected fibers, then the fibers of $f$ foliate $M$. This is called a simple foliation of $M$ and the leaf space can be identified with $N$. Suppose that a ...

**1**

vote

**1**answer

129 views

### Topology of surfaces and mean curvature

The Gauss-Bonnet theorem characterizes topology of surfaces by their Gaussian curvature.
Do there exist results characterizing topology of surfaces embedded in $\mathbb{R}^3$ by their mean ...

**0**

votes

**0**answers

43 views

### Characterization of global sections (which are not products) of a sheaf which is locally a product

In order to compute certain group cohomology sets I have come upon a construction which seems rather general concerning sheaves which are locally products. So I will state the problem here in a ...

**4**

votes

**2**answers

123 views

### functions which covers(good covers) manifolds

Let $M$ be a (not necessarily compact)) smooth manifold.
1.Is there a smooth map $f:M\to \mathbb{R}$ and an open covering $\mathbb{R}=\cup U_{\alpha}$ such that each $f^{-1}(U_{\alpha})$ ...

**27**

votes

**0**answers

512 views

### Are there periodicity phenomena in manifold topology with odd period?

The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...

**6**

votes

**0**answers

246 views

### Classifying space of the higher-structure diffeomorphism group

There is a higher extension of the classifying space $B \mathrm{Diff}$ of the diffeomorphism group implicit in the (infinity,n)-category of cobordisms with (X,zeta)-structure ...

**6**

votes

**2**answers

429 views

### Reference for a fact (?) on homeomorphic knot complements

Does somebody have a reference (or an argument why it should be true) for the following statement?
“Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 ...

**11**

votes

**1**answer

269 views

### Diffeomorphisms and homotopy equivalences sliced over BO(n)

There are some classical results stating sufficient conditions on a manifold $\Sigma$ such that every homotopy equivalence $\Pi(\Sigma) \stackrel{\simeq}{\longrightarrow} \Pi(\Sigma)$ is homotopic to ...

**0**

votes

**0**answers

116 views

### Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17)
\begin{eqnarray*}
...

**0**

votes

**1**answer

93 views

### About the regularity of the boundary of a set

Consider $\Omega_1, \Omega_2$ two convex, bounded domains in $R^n$ with $\Omega_1 \supset \overline{\Omega_2}$ and suppose that $\operatorname{dist}(x, \partial \Omega_2) =\min_{y \in \partial ...

**14**

votes

**5**answers

560 views

### Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting?

First, we make the following observation: let $X: M \rightarrow TM $ be a vector
field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e.
$$ i_X \circ i_{X} ...

**1**

vote

**1**answer

79 views

### Topological description of the regular values of a differentiable function

Is there some kind of description of the set of regular values of a differentiable function $f:\mathbb{R}^{n} \to \mathbb{R}^{m}$ in topological terms?
In particular, is the set of regular values ...

**5**

votes

**1**answer

134 views

### Ehresmann's fibration theorem in the C1 class

I have seen on the French Wikipedia that Ehresmann's fibration theorem is stated with the assumption that everything is $C^2$, see Théorème de Ehresmann. (On the English Wikipedia, the assumption is ...

**7**

votes

**0**answers

116 views

### Topological restrictions from mean curvature bounds

Alexandrov's Theorem says that a compact constant mean curvature hypersurface embedded in $\mathbb{R}^{n+1}$ must be a round sphere.
What happens when the mean curvature is small, or bounded? (For ...

**1**

vote

**0**answers

84 views

### Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here:
...

**8**

votes

**2**answers

544 views

### Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?

The group $\mathbb{Z}_2$ acts on $S^7\times S^7$ by switching the coordinates with fixed point set $\Delta(S^7\times S^7)\cong S^7$. I want to know whether there are some other $\mathbb{Z}_2$ actions ...

**1**

vote

**1**answer

104 views

### Classifying tensor powers of tangent bundles as pullbacks of appropriate tangent bundles

Assume we are given a smooth manifold $M$ and let $TM^{\otimes r}, r>1,$ be some
tensor power of its tangent bundle.
Is there any general observation/result
saying when there exists a manifold ...

**13**

votes

**1**answer

347 views

### Higher Cerf Theory

Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that ...

**9**

votes

**0**answers

115 views

### Detecting invertible elements in group rings by their images for finite quotients of the group

Let $G$ be a "nice" infinite group: at least finitely presented and residually finite, maybe also linear and right-orderable (or even bi-orderable, or residually free nilpotent).
Consider an element ...

**6**

votes

**0**answers

85 views

### Can one detect smoothness of $k$-forms with $k$-dimensional manifolds

Fix an integer $k\ge 1$. Let $X$ be a smooth manifold. It is well-known, that a real valued function on $X$ is smooth if its restriction along all smooth maps $M\to X$ for manifolds of dimension $\le ...

**3**

votes

**1**answer

82 views

### cartesian product rigidity for the punctured open disc

Q1: Let $D^n$ ($n\geq 1$) be the n-dimensional open disk. If $D^n-\{0\}$ is homeomorphic to $X\times (0,1)$, for some topological space $X$, does it necessarily follow that $X$ is homeomorphic to ...

**6**

votes

**1**answer

240 views

### Immersion of $S^1$ in $\mathbb{R}^2$ that can be extended to $\mathbb{D}$

I was wondering about $\mathcal{C}^{\infty}$ immersions $S^1 \longrightarrow \mathbb{R}^2$ which are the restriction to $\partial \mathbb{D}$ of an immersion $\overline{\mathbb{D}} \longrightarrow ...