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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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        64 views

        Extension of Vector Field in the $\mathcal{C}^r$ topology

        This question was previously posted on MSE. Let $M\subset \mathbb{R}^n$ be a compact smooth manifold embedded in $\mathbb{R}^n$, we define $$\mathfrak{X}(M) := \{X: M \to \mathbb{R}^n;\ X\mbox{ is ...
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        56 views

        Globalising fibrations by schedules

        In a paper in Fund Math 130.2 (1988): 125-136. http://eudml.org/doc/211719 Dyer and Eilenberg give an account of the local-global theorem for fibrations by proving a "Schedule Theorem" that, given a ...
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        2answers
        674 views

        Examples of odd-dimensional manifolds that do not admit contact structure

        I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure. Can someone provide me with some examples?
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        9 views

        Boundary value problems for differential inclusions with fractional order

        I'm having technical problems or just lack of knowledge problems, so I would appreciate your help. the problem:: let $v_{*}\in F$, and for every $w \in F$, we have $$|v_{n}-v_{*}| \leq |v_{n}-w|+|w-...
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        1answer
        93 views

        Classification of all equivariant structure on the Möbius line bundles

        Is there a classification of all equivariant structures of the M?bius line bundle $\ell\to S^1$?. For example the antipodal action of $\mathbb{Z}/2\mathbb{Z}$ on $S^1$ cannot be lifted to the total ...
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        180 views

        Do smooth manifolds admit unique cubical structures?

        It seems to me that a smooth manifold should admit the structure of a cubical complex by Morse theory, since handle attachments seem to be perfectly cubical maps. Is this cubical structure "...
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        2answers
        305 views

        Is this a submanifold?

        Let $(M,g)$ be a compact Riemannian manifold with an isometric action $\rho : G \to \mathrm{Iso}(M)$ by a compact Lie group $G$. There is a natural extension of $\rho$ to $TM$ given by: $$\psi : G \...
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        1answer
        286 views

        Smooth vector fields on a surface modulo diffeomorphisms

        Let $\Sigma$ be a two-dimensional connected smooth manifold without boundary. (Feel free to assume it is compact and orientable.) Let $\mathcal{X}(\Sigma)$ denote the smooth vector fields on $\Sigma$...
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        1answer
        124 views

        $L^{2}$ Betti number

        Let $\tilde{X}$ be a non-compact oriented, Riemannian manifold adimits a smooth metric $\tilde{g}$ on which a discrete group $\Gamma$ of orientation-preserving isometrics acts freely so that the ...
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        On the homotopy type of $\mathrm{Diff}(\mathbb{S}^3)$

        I am confused with the following argument. I know I am doing something wrong but I can't find my mistake. On one hand, one knows that if $M$ is a Lie group, then $$\mathrm{Diff}(M)\simeq M\times\...
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        1answer
        127 views

        Homology of universal abelian cover of a manifold

        If one define the universal abelian covering $M_0$ of a manifold $M$ as the abelian covering (i.e. normal covering with abelian group of deck transformations) that covers any other abelian covering, ...
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        54 views

        Symplectic vector fields everywhere transverse to a co-dimension one hypersurface

        Usually when speaking about vector fields transverse to a hypersurface in a symplectic manifold, we talk about Liouville vector fields, i.e. vector fields $X$ with the property that $\mathcal{L}_X\...
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        votes
        0answers
        191 views

        What mathematical background to i need in order to understand proofs of the h-cobordism theorem?

        I am about to finish my undergraduate studies and I really enjoyed the topology and differential-geometry classes. I'd love to continue studying differentialtopology and i considered doing some ...
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        0answers
        62 views

        Transitivity of Diff on the space of embeddings of balls

        Given 2 symplectic embeddings $g_0$ and $g_1$ of a 4-ball of radius $r \leq 1$ into the 4-ball of radius 1 (all equipped with the standard symplectic form coming from $\mathbb{R}^4$), does there exist ...
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        1answer
        235 views

        Different definitions of the linking number

        Assume that $$ \iota_1:\mathbb{S}^k\to\mathbb{R}^n, \quad \iota_2:\mathbb{S}^\ell\to\mathbb{R}^n, \quad k+\ell=n-1, $$ are two embeddings of spheres with disjoint images $\iota_1(\mathbb{S}^k)\cap\...

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