**1**

vote

**0**answers

40 views

### Complexity of mappings (forms) in R. Thom's “Structural stability and morphogenesis”

In his "Structural stability and morphogenesis", R. Thom (especially in the chapter about dynamics of forms) among other things speculates about a notion of complexity of a "form" (mapping between ...

**0**

votes

**0**answers

89 views

### book suggestion in fourier analysis and differential topology [on hold]

I am at present following Vinberg and Onischik's Lie Groups and Algebraic Groups and I find it a fantastic book with theory developed through a series of problems left to the reader. I would like to ...

**6**

votes

**2**answers

129 views

### Topological relationships between family of transversal intersections of manifolds

Let $M$ and $N$ be submanifolds of $\mathbb{R}^n$ and let $a(t)$ be a smooth path in $\mathbb{R}^n$ such that $M+a(t)$ intersects $N$ transversally for all $t \in [0,1]$. Is there a nice relationship ...

**-1**

votes

**1**answer

71 views

### Generalized Leibniz rule [closed]

Suppose on a manifold $M$ we have a differential operator $D$ of order 1 from the smooth sections $C^\infty(M, E)$ to smooth sections $C^\infty(M, F)$, where $E$ and $F$ are vector bundles of rank $n$ ...

**2**

votes

**1**answer

119 views

### Extending diffeomorphisms of some faces of the standard $n$- simplex to a diffeomorphism of the $n$-simplex

I would be grateful if you could help me with the following question:
Let $n\in \mathbb{N}$ and $k\in \{1, \ldots, n+1\}$. For every $i=1, ..., k$ let $f_i : \partial_i \Delta^n \rightarrow ...

**5**

votes

**1**answer

180 views

### Analytic maps in homotopy classes

Let $\mathcal M$ be a compact connected real-analytic manifold. It is well known that every continuous map $f\colon\mathcal M\to\mathbb S^1$ is homotopic to a smooth map. My question is the following. ...

**0**

votes

**0**answers

33 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 ...

**0**

votes

**0**answers

21 views

### Continuousness Property of the Map Space $\mathcal{M}(W,M)$

If $W$ and $M$ are manifolds we write $\mathcal{M}(W,M)$ for the set of differential maps of $W$ into $M$.The set is equipped with the $C^{r}-$ topology for some fixed $r$ with $1\leqslant ...

**3**

votes

**1**answer

77 views

### A reference for an equivariant Morse Lemma

Does anybody knows a reference for the following statement?
Let $S^1$ acts on $\mathbb{C}^n$ in the usual (diagonal) way and $f:\mathbb{C}^n\to\mathbb{R}$ a smooth $S^1$-invariant function defined in ...

**2**

votes

**3**answers

121 views

### Frechet Derivative in General Topological Vector Space

If I have a two Hausdorff topological vector spaces, $E$ and $F$ and a mapping $f:E\to F$, is it possible to have a meaningful notion of the derivative of $f$ if the space cannot be endowed with a ...

**3**

votes

**0**answers

134 views

### Boundary of an open, bounded and convex set in $\mathbb{R} ^n$

Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...

**1**

vote

**0**answers

64 views

### restriction to the boundary in Morse theory

Given a compact manifold with boundary $(M,\partial M)$. There is a natural pullback map
$$ H^*(M) \to H^*(\partial M) $$
I'm wondering if there is a reference that:
1) constructs this map in ...

**2**

votes

**1**answer

119 views

### local approximation of a vector field on a Riemannian manifold

Let $(M^n,g)$ be a Riemannian manifold, and let $V$ be a $C^{\infty}$ vector field on $M$. Is it possible to locally approximate $V$ by gradient vector fields $\nabla f_i$, such that the ...

**4**

votes

**2**answers

177 views

### Question about lower homology class of cobordism

Assume there are three differential oriented manifold $M_0$, $M_1$, $W$ with $\partial W= M_0 \coprod -M_1$. Denote dim $M_0$=dim $M_1$=n, and dim W=n+1.
We know that for the highest homology class, ...

**4**

votes

**1**answer

159 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 ...

**0**

votes

**1**answer

144 views

### determinant of integrals of forms

Let $A$ be a complex abelian variety of dimension $d$. Let $\omega_1, \ldots, \omega_j \in H^0(A, \Omega^1_A)$ be linearly independent (so $j \leq d$) and consider $\gamma_1, \ldots, \gamma_j \in ...

**10**

votes

**3**answers

710 views

### Which submanifolds are zero sets of $\mathbb{R}^n$-valued maps?

If $M$ is a smooth, compact, orientable manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal ...

**4**

votes

**1**answer

464 views

### The space of diffeomorphisms on a manifold

It is well known that given a compact connected smooth manifold without boundary $M$, the set of diffeomorphisms $Diff^{r}(M)$ of $M$ for $r≥1$, is open in $C^{r}(M)$, the set of continuous functions ...

**1**

vote

**0**answers

41 views

### Topology of fibers of operators under C^{\infty} convergence

A smooth family of maps $f_t : L^2 (\mathbb{R}) \rightarrow \mathbb{R}^{n}$ is given, with $t \in (0,1]$. Suppose that when $t \rightarrow 0$ the family restricted to balls of given radius converges ...

**3**

votes

**2**answers

139 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

**2**answers

201 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 ...

**6**

votes

**0**answers

71 views

### What is the space of pairwise distances associated with $n$-tuples of points on a circle?

Let $\mathbb{T}^n$ be the $n$-dimensional torus with coordinates $\mathbf{z}=(z_1,\dots,z_n)$ (so that each $z_i\in[-\pi,\pi)$ is a coordinate on a one-dimensional circle). For each ...

**14**

votes

**3**answers

290 views

### Existence of sections of the evaluation map for the diffeomorphism group

Let $M$ be a closed connected oriented smooth manifold and $\mathrm{Diff}_{+}(M)$ the group of orientation preserving diffeomorphisms of $M$ endowed with the compact-open topology. Pick a base point ...

**1**

vote

**0**answers

46 views

### Question about embeddings of connected sum of manifolds [duplicate]

If $M_j,j=1,2$are smooth manifolds and we have two embeddings $i_j:M_j\to\mathbf{R}^k(j=1,2)$(with $k$ fixed). How can we construct a embedding of the connected sum $M_1\sharp M_2$ into ...

**1**

vote

**1**answer

77 views

### Invariant of isotopy of curves in a surface.

Suppose $S_g$ is a sorface of genus $g>1$. Let $\gamma_1$ and $\gamma_2$ be two simple closed curves containing points $p_1, p_2$. Suppose $\gamma_1$ and $\gamma_2$ are isotopic. Now there can be ...

**0**

votes

**0**answers

93 views

### The Jordan-Brouwer Separation Theorem for Manifold

I have read the paper The Jordan-Brouwer Seperation Theorem written by Wolfgang Schmaltz. The main result in paper is below
Any compact, connected hypersurface $X$ in $\mathbb R^n$ will divide ...

**2**

votes

**1**answer

121 views

### $C^1$ stability conjecture on non-compact manifolds

In the study of dynamical systems, the link between structural stability and Smale's Axiom A has been explored by many authors. One of the important outcomes of this endeavor was the proof of $C^1$ ...

**0**

votes

**0**answers

46 views

### Total differential of Lipschitz submanifolds embedding

My interest is analysis on Lipschitz manifolds. I want to define traces of differential forms on a Lipschitz submanifold $N$ of a Lipschitz manifold $M$. In other words, I want to push-forward ...

**2**

votes

**1**answer

79 views

### Genericity of maps which are transverse when restricted to a submanifold

Let $M$ and $N$ be smooth manifolds, and $A\subset M$, $B\subset N$ be smooth embedded submanifolds. I am looking for a reference for a theorem on the following lines:
The set of smooth maps $h\in ...

**9**

votes

**1**answer

281 views

### How to flow submanifolds?

Motivation
We want a consistent way of perturbing a submanifold away from itself. For $0$-dimensional submanifolds, this is the same data as a nowhere-vanishing vector field: we may flow the points ...

**15**

votes

**1**answer

338 views

### Idempotents split in category of smooth manifolds?

In his paper "Qualitative Distinctions Between Some Toposes of Generalized Graphs" (reproduced here), page 267, Lawvere says that the idempotent-splitting completion of the category of open sets of ...

**1**

vote

**0**answers

153 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 ...

**4**

votes

**0**answers

224 views

### Hirzebruch's ICM talk

Is there an English translation of Hirzebruch's 1958 ICM lecture note:
KOMPLEXE MANNIGFALTIGKEITEN
Thank you very much!

**1**

vote

**0**answers

100 views

### Intermediate value theorem for the Jacobian determinant restricted to a curve

Let $f:R^N \to R^N$ be a differentiable mapping, and $J_f$ its Jacobian determinant. Suppose that $\exists a,b \in R^N : J_f(a)<0,J_f(b)>0$. Is it right that on every continous curve connecting ...

**4**

votes

**2**answers

210 views

### Rank of a jet bundle of a vector bundle. Interpretation of the first jet bundle

I am trying to understand the jet bundles but currently I am stuck on the following questions:
Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...

**2**

votes

**0**answers

101 views

### Logarithmic de Rham complex induced map on Hypercohomology

• Question: Assume X
is an algebraic manifold over $\mathbb{C}$
and $D$
is a simple normal crossings divisor. The logarithmic de Rham complex $\left(\Omega_{X}^{\bullet}\left(log\ ...

**1**

vote

**1**answer

184 views

### How to classify continuous/differentiable maps from $T^2$ to $U(N)$?

I read a physics paper, of which the main idea is based on the topological classification of maps from 3-torus to the space of $N\times N$ unitary matrices. To quote their equation (4), which gives a ...

**4**

votes

**1**answer

184 views

### Exercises around Diffeological Spaces or a Diffeologic Atlas Theory

Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...

**10**

votes

**0**answers

176 views

### A simple proof that parallelizable oriented closed manifolds are oriented boundary?

So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...

**1**

vote

**2**answers

151 views

### Smooth but non-analytic kernel functions

Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?

**1**

vote

**0**answers

69 views

### Boundary of fibers of submersions

Let $M$ be a smooth manifold with boundary (say of dimension $m$) and let $N$ be a smooth manifold with no boundary (say of dimension $n$ with $m\geq n$). We have the following classical result:
...

**9**

votes

**0**answers

249 views

### Topological type of Brieskorn manifolds

Let us consider the complex hypersurface and suppose that $n\geq 3$:
$$F(d,n)=\{(z_0,\ldots,z_n)\in \mathbb{C}^{n+1}:z_0^d+z_1^d+\ldots+z_n^d=0\}$$
and the link $V(d,n)=F(d,n)\cap S^{2n+1}_{\epsilon}$ ...

**3**

votes

**2**answers

148 views

### Markov Partitions for toral automorphisms

I know that my question is more practical than theoretical. But, I do know where to look for the theoretical sources.
I want to find a program in the case that it exists (does it?), or to program it. ...

**0**

votes

**0**answers

50 views

### Straightening the level sets of a smooth function

Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ be a smooth function and
$D\subset\mathbb{R}^{n}$ be a closed ball such that $f$ has no critical points
on $D$: $\nabla f(x)\neq0$ for $x\in D$.
My ...

**8**

votes

**2**answers

225 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 ...

**-3**

votes

**2**answers

129 views

### The boundary of this set is piecewise smooth? [closed]

Consider a sequence of open sets in $R^n$: $\Omega_1 \supset \Omega_2 \supset\cdots$. Consider that this sets are bounded, convex with the boundary piecewise smooth .When i say smooth i mean ...

**5**

votes

**2**answers

275 views

### A good metric for transversal intersections

Let $V_1,\ldots,V_k$ be a transversal set of smooth compact orientable sub-manifolds of a compact orientable manifold $M$, and set $V=\bigcap V_i$.
Is it always possible to equip a neighborhood $U$ ...

**1**

vote

**0**answers

49 views

### Twisted calibrations and sufficient conditions on homology of sub-manifolds

I think my question is somehow easy to solve, but I'm not very familiar with algebraic topology, so I'm not able to figure out the solution for myself. I'm working on a problem in metric geometry and ...

**2**

votes

**0**answers

198 views

### Invariants of solutions of systems of equations

What can be said about invariants of zero set of a function that don't change under small enough continuous perturbations of the functions? I define an $\epsilon$-perturbation $g$ of $f$ to be any ...

**-1**

votes

**2**answers

163 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 $M$?