# Questions tagged [differential-topology]

The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.

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### Separating two submanifolds [on hold]

Let $M, N, X$ are compact manifolds. Let $f_1:M \rightarrow X$ and $g_1: N \rightarrow X$ be any two embeddings. Is it always possible to find embeddings $f_2 $ homotopic to $f_1$ and $g_2$ homotopic ...

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### Can we approximate elements in $L^2$ via smooth maps while preserving pointwise orthogonality constraints?

I have the following problem:
Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Suppose we are given an open subset $U \subseteq \mathbb{D}^n$ of full measure in $\...

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### Quotient of arbitrary free involution on $S^n$

If we consider arbitrary free involution on $S^n$, then the quotient need not be diffeomorphic to $\Bbb RP^n$ if $n\geq 5$ and a reference for this is "some curious involutions of spheres"
by Morris W....

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### What finite simple groups appear as factors of surface fundamental groups?

Let $\Sigma_g$ be the a closed orientable surface of genus $g$.
My somewhat naive question: what is known about simple finite factors of $\pi_1(\Sigma_g)?$ In particular, I know that the composition ...

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### A relaxed form of quantitative transversality

Let $f$ be a holomorphic function on the unit ball $B_1(0) \subseteq \mathbb{C}^n$. Then given any pair of constants $\delta, \epsilon > 0$, does there exist a smooth function $g: B_1(0) \to \...

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### Smooth structure for probability measures on separable Hilbert space

If $H$ is a separable Hilbert space (say, $L^2(\mathbb{R}^d)$ for concreteness), then it is well-known that the unit sphere $S$ of $H$ is a Hilbert manifold modeled on $H$ itself. The coordinate ...

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### Perfect groups of infinite order

Let $M$ be a closed, minimal, hypersurface in the sphere $S^{n-1}$, $n\geq 4$. Suppose $M$ has $H^1(M,\mathbb{Z})=0$. What can we say about the cardinality of the first fundamental group of $M$, $\...

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### Path of Diffeomorphisms Fixing the Boundary

Could you please let me know if the following is true. The problem came up while constructing a solution of a PDE. I have browsed through the net for an answer. While I came across some articles ...

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### Diffeomorphism type of the added sphere in simply connected surgery

A classical result of simply connected surgery theory, is that if two normal maps $f:M_i\rightarrow X$, $i=0,1$ are normally cobordant and if the dimension of the manifolds is odd, there exists a ...

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### Oriented vector bundle with odd-dimensional fibers

Is it true that for every oriented vector bundle with odd-dimensional fibers, there is always a global section that vanishes nowhere?

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### Existence of topologically mixing (discrete) dynamical system on manifold

If $M$ is a connected $(d\geq 2)$-dimensional smooth closed manifold, then does there exist a class $C^1$-diffeomorphism $\phi$ from $M$ onto itself, such that $(M,\phi)$ is a topologically mixing (...

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843 views

### Link of a singularity

I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$.
If we set $x = x_1+ix_2, y = y_1+iy_2, z ...

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### Star-shaped domain in $\mathbb{C}P^2$

Consider $(\mathbb{C}P^2,\omega_{FS})$ where $\omega_{FS}$ is the standard Fubini-Study form. Let $L$ denote a sphere in $\mathbb{C}P^2$ in the class $\mathbb{C}P^1$. Further let $\int_{L} \omega_{FS} ...

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### Wildness of codimension 1 submanifolds of euclidean space

This question arose out of this stack exchange post. I am wirting a thesis about the $s$-cobordism theorem and Siebenmann's work about end obstructions. Combined they give a quick proof of the ...

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### The space of $k$ differential forms as a Fréchet space

Given a smooth manifold $M$, can define define seminorms on $\Gamma(U,\bigwedge^kT^{\ast}M)$ for $U$ a coordinate open set by the following: $p^{s}_L(u = \sum_{I}u_I dx_I) = \sup_{x \in M}\max_{|I|=p, ...