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      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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      60 views

      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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      64 views

      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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      183 views

      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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      207 views

      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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      53 views

      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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      79 views

      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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      1answer
      156 views

      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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      1answer
      120 views

      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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      1answer
      320 views

      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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      1answer
      51 views

      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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      80 views

      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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      3answers
      282 views

      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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      85 views

      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, ...

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