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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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        votes
        0answers
        172 views

        3-fold of general type homeomorphic to rational 3-fold

        Is there a smooth (complex projective) 3-fold of general type which is homeomorphic (in the complex topology) to a rational $3$-fold? I am aware of such examples in complex dimension $2$, for ...
        3
        votes
        0answers
        247 views

        Topological approach to create a space between clouds

        I have a dataset associated with labels. According to https://arxiv.org/pdf/1802.03426.pdf --> UMAP (Uniform Manifold Approximation and Projection) which is a novel manifold learning technique for ...
        2
        votes
        1answer
        134 views

        What is the topological/smooth analogue of Nagata compactification

        A celebrated theorem of Nagata and subsequent refinements to schemes and algebraic spaces say that over a not-completely-monstrous base scheme, any separated morphism can be openly immersed in a ...
        4
        votes
        3answers
        254 views

        Elliptic regularity on compact manifold without boundary

        Let $(M,g)$ be a Riemannian compact manifold without boundary, and $\Delta$ is the Laplace-Beltrami operator on $M$. Is there any result on the elliptic regularity like this: For any $u\in H^1(M)$, ...
        6
        votes
        1answer
        186 views

        Action of diffeomorphism group on non-vanishing vector fields

        Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0)...
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        votes
        1answer
        114 views

        Intersection of hyperspheres

        Suppose we have $n$ hyperspheres in $\mathbb{R}^m$, $m\geq n$, of centers $x_1,\ldots x_n$, $x_i\neq x_j\,\forall i,j$, and radii $r_1,\ldots ,r_n$. Suppose that, for every $i,j$, the quantities $r_i$,...
        5
        votes
        0answers
        84 views

        Induced new structures on Poincare dual manifolds

        "R. C. Kirby and L. R. Taylor, Pin structures on low-dimensional manifolds (1990)" shows Given a spin structure on $M^3$, the submanifold $\text{PD}(a)$ can be given a natural induced $\text{Pin}^-$...
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        votes
        0answers
        145 views

        GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?

        GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...
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        votes
        0answers
        44 views

        Relation between the orientation sheaves of the interior and the boundary of a topological manifold

        Let $(M, \partial M)$ be an $n$-dimensional topological manifold with boundary. Let $\mathcal{O}_{M \setminus \partial M}$ and $\mathcal{O}_{\partial M}$ denote the orientation sheaves of $M \setminus ...
        5
        votes
        1answer
        126 views

        Relating bordism groups of different dimensions

        Let $M_d$ be a $d$-manifold generator of a subgroup of bordism group $$ \Omega_d^{G}, $$ or further generalization $$ \Omega_d^{G}(K(\mathcal{G},n+1)), $$ which $G$ is the given structure ...
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        votes
        3answers
        899 views

        Is there a closed non-smoothable 4-manifold with zero Euler characteristic?

        I will just repeat the title: Is there a closed non-smoothable 4-manifold with zero Euler characteristic? I am guessing yes simply based on other existence theorems I have seen for 4-manifolds.
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        vote
        0answers
        43 views

        Upper bounds on $\epsilon$-covers of arbitrary compact manifolds

        Let $M \subset \mathbb{R}^d$ be a compact $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every ...
        4
        votes
        0answers
        89 views

        Lower bound on $\epsilon$-covers of arbitrary manifolds

        Let $M \subset \mathbb{R}^d$ be a $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every point $...
        5
        votes
        2answers
        498 views

        Classification of closed 3-manifolds with finite first homology group?

        I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H_1(M)$. Since $H_1(M)$ is the abelization of the fundamental group $\pi_1(M)$, ...
        6
        votes
        1answer
        329 views

        Manifolds with negative dimension – Definition, References

        Does the concept of differential manifold with negative dimension make sense, in differential geometry? If yes, how is it defined? Do you have any reference to recommend? My problem was born in ...

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