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        Questions tagged [manifolds]

        A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n.

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        1answer
        54 views

        How to show $UM$ is an embedded $(2m-1)$-dimensional submanifold of $T\Bbb R^n\approx \Bbb R^n\times \Bbb R^n$? [on hold]

        Because $TM$ is an embedded $2m$-dimensional submanifold of $T\Bbb R^n\approx \Bbb R^n\times \Bbb R^n$, we can define the smooth map $\varphi: TM\to \Bbb R$ by $(x_1,\cdots,x_n,v_1,\cdots,v_n)\...
        4
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        1answer
        184 views

        Existence of universal arrow from manifolds to forgetful functor of Lie groups

        Let $M$ be a manifold, and $U$ be the forgetful functor from the category of Lie groups to the category of manifolds. My question is whether there is a universal arrow $(G, i)$ from $M$ to $U$? More ...
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        2answers
        206 views

        Line bundles trivial outside of codimension 3

        Let $X$ be a CW complex (possibly a topological/smooth manifold) of dimension $n$, $L\to X$ a complex line bundle and $Y\subset X$ a subcomplex (possibly a submanifold) contained in the codimension 3 ...
        2
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        0answers
        132 views

        Category of Manifolds and Maps: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF? [closed]

        Please let me denote the following (TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold (PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF (PL) ...
        6
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        1answer
        703 views

        Cobordism Theory of Topological Manifolds

        Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds. Cobordism Theory for DIFF/Differentiable/smooth manifolds However, there are ...
        3
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        104 views

        Naturality of Poincaré–Lefschetz

        Let $X$ be compact and Hausdorff, $A\subseteq B\subseteq X$ both closed such that $X\setminus A$ is an open orientable $d$-manifold. Then also $X\setminus B$ is an open orientable $d$-manifold. We ...
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        0answers
        69 views

        How do these topological results imply the inverse function theorem?

        In this MO question, Terrence Tao inquires about the everywhere differentiable inverse function theorem. This answer claims the theorem may be deduce from fairly intricate topological results of ...
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        0answers
        72 views

        “Smooth” Serre Fibrations (?)

        Let $M,N$ be manifolds, $f:M \to N$ be a map. In order to understand if $f$ is a serre fibration, it is enough to test it against differntiable maps $I^p \to M, I^{p+1} \to N$? What about smooth maps?...
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        123 views

        Counting fixed points for Hamiltonian symplectomorphisms on $T^{2}$

        This question is motivated by the Lorenz curve used in economic analysis and also the Penrose diagram used in general relativity, used by physicists in order to visualise causal relationships in ...
        2
        votes
        0answers
        79 views

        How much is the theory of piecewise-smooth manifolds different from the PL and smooth ones?

        There are many differences between the smooth manifolds category and piecewise linear manifolds category when it comes to classification, embedding of these manifolds inside each other or otherwise. ...
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        75 views

        Notion of knotting for a compact $n$-manifold inside an $m$-manifold (with the constraint $n=m?2$)

        What is the notion of knotting for a compact $n$-manifold inside an $m$-manifold?
        3
        votes
        1answer
        90 views

        Unknotting a compact manifold in the PL setting

        The general position theorem asserts that any $M$ $m$-manifold unknots in $R^n$ provided $n\geq 2m+2$. The general position theorem assumes a smooth setting. Is unknotting still hold in the PL setting?...
        4
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        2answers
        202 views

        Notion of linking between two general $p$ and $q$ manifolds embedded in a higher dimensional manifold

        Let $M$, $N$ be compact, connected, oriented manifolds without boundary embedded in $\mathbb{R}^m$ of dimensions $p$ and $q$ respectively. I know that when $m=p+q+1$ we can define the linking between $...
        5
        votes
        2answers
        316 views

        Transitive embedding of the projective plane $\Bbb R P^2$ into the $4$-sphere

        Is there an embedding (i.e. injective continuous map) $$\phi:\Bbb R P^2\hookrightarrow S^4\subseteq\Bbb R^5$$ of the projective plane $\Bbb R P^2$ into the $4$-sphere, that is transitive, i.e. ...
        21
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        1answer
        553 views

        Closed manifold with non-vanishing homotopy groups and vanishing homology groups

        Is there a closed connected $n$-dimensional topological manifold $M$ ($n\geq 2$) such that $\pi_i(M)\neq 0$ for all $i>0$ and $H_i(M, \mathbb{Z})=0$ for $i\neq 0$, $n$? The manifold $S^1\times S^2$ ...

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