# Tagged Questions

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### Branch point and alexandrov embeddedness

This is a question I have asked on mathstackexchange with a bounty but without any answer; it is probably more adapted to mathoverflow: Let us assume that $\Sigma_n$ is a sequence of topological ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 $U$ ...
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### 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 $M$?
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### Topological invariants of toroidal orbifolds

Which are the most powerful topological invariants of toroidal orbifolds? In particular I am looking for topological invariants of two-dimensional toroidal orbifolds such as $T^{2}/Z_{k}\times Z_{k}$ ...
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### 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 $x\in D^m$ ...
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### Finiteness of De Rham cohomology of smooth quasi-projective varieties

Let $U$ be a smooth quasi-projective variety over $\mathbf{C}$. Let $U^{\infty}$ be $U$ but thought of as a smooth manifold. Q1: Is there a simple proof (so it should avoid Hironaka's ...
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### 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 ...
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### Embeddings of complex projective Stiefel manifolds

Let $k\leq n$ and define \begin{align*} V_k(\mathbb{C}^n) & =\{\textrm{orthogonal $k$-frames in } \mathbb{C}^n\}\\ & = \{A\in \textrm{Mat}_{n\times k}{(\mathbb{C})}\mid A^\ast A = {1}_k\} ...
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### What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...
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### 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 ...
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### 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 $\partial X = M$ ...
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### even dimensional manifold homotopic to a symplectic or complex manifold

I have the following question: Let $M$ be an even dimensional Riemannian manifold. Under which conditions does there exists a homotopy to some symplectic manifold? is there any chance that such a ...
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### Extension of diffeomorphisms preserving bilateral bounds of the derivatives

Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, which is a diffeomorphism from the domain to its image, with the upper and lower ...
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### Understanding maps from M to R^n, for n>dim M

I am interested in "approximating" smooth maps from a compact smooth manifold $M$ of dim $m$ into $\mathbb R^n,$ for $n>m,$ by "nice" maps, with properties similar to those of Morse functions. Of ...
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### 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 ...
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### Does Frobenius theorem apply to vector-valued function?

We know Frobenius theorem handle pde systems like $\{Xf=0, Yf=0\}$ requiring Lie bracket $[X,Y]\equiv 0 \mod X, Y$ for completely integrability of the system. However, how to handle systems like ...
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### Fatou sets and topological entropy

Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...
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### Exotic 4-manifolds with even positive partial betti number

It seems that usually smooth structures on compact 4-manifolds are distinguished by Seiberg-Witten/Donaldson invariants. And I don't know another direct way to do that. But in the case when $b_2^+$ is ...
Let $M$ be a smooth manifold. If $X$ and $Y$ are two Whitney objects, i.e. subsets with a given Whitney stratification, then $X$ and $Y$ are transverse if each stratum of $X$ is transverse to each ...