**2**

votes

**0**answers

63 views

### Brouwer fixed points via flow

Let $B$ be a closed ball in $n$-dimensional space, let $f$ be a "sufficiently smooth" map from $B$ to $B$, and let $p$ be a point on the boundary of $B$.
It seems to me that something like the ...

**5**

votes

**2**answers

211 views

### What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...

**2**

votes

**0**answers

45 views

### How to prove that all smooth vector bundles on a given vector bundle are the pull back of a vector bundle on the base [migrated]

Recently, during a conversation, I heard about the result (previously mentioned also here on MO), whose statement is reported below. Not having the specific background necessary to reconstruct a proof ...

**3**

votes

**0**answers

192 views

### A lifting problem

Let $E\overset{\pi'}{\longrightarrow} B'$ and $E\overset{\pi}{\longrightarrow} B$ be vector bundles.
For $i=0,1$, let $f_i$ be a fiber-preserving open embeddings of $\pi'$ into $\pi$, with $g_i$ the ...

**0**

votes

**0**answers

48 views

### Geometric Proof of the Hirzebruch Signature Theorem (i.e. using Connections)

Is there a geometric proof of the Hirzebruch Signature Theorem, that is to say: Is the a proof that does not use the topological definition of the Chern/Pontrijagin classes, but only the Chern--Weil ...

**1**

vote

**0**answers

81 views

### Thom-Pontryagin construction for pairs

Is there a generalization of the Thom-Pontryagin construction in the following sense? Let $M$ be a smooth manifold, $\partial M=A\cup B$ where $A$ and $B$ are $m-1$-manifolds with a common boundary ...

**12**

votes

**0**answers

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 ...

**12**

votes

**2**answers

513 views

### Does every compact manifold exhibit an almost global chart

Let $M$ be a compact connected manifold.
Is there a chart $\Psi:U \to \mathbb{R}^n$ such that the closure of $U$ is $M$?
This is true for $S^n, T^n, K$, all compact surfaces, etc.
If it is not true in ...

**1**

vote

**1**answer

68 views

### Connected representant of a framed cobordism class (reference needed)

Let $N^n\subseteq M^m$ be a submanifold with a framing of the normal bundle, $2n<m$. Then $N^n$ is framed cobordant (in $M^m$) to something connected.
I believe it could be proved by directly ...

**5**

votes

**0**answers

189 views

### Are inclusions from spaces of $C^\infty$ sections into spaces of $C^k$ sections homotopy equivalences?

[EDIT: The answer to my original question was obviously no, as user56365 pointed out. Here is what I should have asked.]
For finite-dimensional smooth manifolds $E,M$, let $E\to M$ be a smooth fibre ...

**4**

votes

**0**answers

82 views

### Connection between framed cobordisms and zero sets

Let $W\subseteq M\times[0,1]$ be a framed submanifold (a framed cobordism in $M$) and $2w<m-1$ where $w,m=\dim W,M$. Assume that $M$ is compact and that $W\cap M\times \{0\}$ is the zero set of a ...

**11**

votes

**0**answers

309 views

### Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle

My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...

**1**

vote

**0**answers

45 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 ...

**6**

votes

**2**answers

137 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

75 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

130 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

185 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

35 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

79 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

126 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

141 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

67 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

122 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

180 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, ...

**6**

votes

**1**answer

192 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

238 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 ...

**11**

votes

**3**answers

736 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

481 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 ...

**2**

votes

**0**answers

42 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

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

**2**answers

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 ...

**6**

votes

**0**answers

73 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

292 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

80 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

104 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

124 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

80 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

285 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

352 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

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 ...

**4**

votes

**0**answers

232 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

104 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

213 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

103 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

187 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

188 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

178 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 ...