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        All Questions

        22
        votes
        1answer
        512 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 $S^1\times S^2$ ...
        4
        votes
        1answer
        242 views

        Is a manifold-with-boundary with given interior and non-empty boundary essentially unique?

        Let $M$ be a compact connected manifold-with-boundary such that $\circ M \neq \emptyset$, where $\circ M$ is the boundary of $M$. Let $N$ be a compact connected manifold-with-boundary such that $\circ ...
        9
        votes
        0answers
        265 views

        History of the definition of smooth manifold with boundary

        I am trying to determine the earliest source for the definition of smooth ($C^\infty$) manifold with boundary. Milnor and Stasheff (1958) give a definition, but a scrutiny of that definition shows it ...
        8
        votes
        0answers
        146 views

        $k$ times differentiable but not $C^k$ manifold

        I asked the following question on Math Stack Exchange 3 months ago but got no answer. So maybe Math Overflow is a more suitable place for such a question: I cannot find the notion of $k$ times ...
        7
        votes
        1answer
        202 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)...
        5
        votes
        0answers
        90 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 $\text{Pin}^-$...
        5
        votes
        1answer
        292 views

        Every unorientable 4-manifold has a $Pin^c$, $Pin^{\tilde c+}$ or $Pin^{\tilde c-}$ Structure

        The precise statement on J. W. Morgan's "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds (MN-44)" that 4-manifold $X$ admits a Spinc structure (Lemma 3.1.2) ...
        4
        votes
        0answers
        179 views

        Non-spin 5-manifold and $2^2$-Bockstein homomorphism

        The $2^2$-Bockstein is $\beta_4$ is associated to $$0\to\mathbb{Z}/2\to\mathbb{Z}/{8}\to\mathbb{Z}/{4}\to 0,$$ (The $2^n$-Bockstein homomorphism $$\beta_{2^n}:H^*(-,\mathbb{Z}/{2^n})\to H^{*+1}(-,\...
        8
        votes
        1answer
        261 views

        Non-triangulable 4-manifold as a boundary of some 5 manifold

        We know that there are non-triangulable 4-manifolds, such as the E$_8$ manifold. Can E$_8$ manifold be a boundary of some 5-manifold $M_5$? Can such a $M_5$ be triangulable or non-triangulable? What ...
        6
        votes
        2answers
        474 views

        Any 3-manifold can be realized as the boundary of a 4-manifold

        We know "Any closed, oriented $3$-manifold $M$ is the boundary of some oriented $4$-manifold $B$." See this post: Elegant proof that any closed, oriented 3-manifold is the boundary of some ...
        11
        votes
        1answer
        383 views

        Is there a PL, or topological, bordism hypothesis?

        The bordism hypothesis says that the $(\infty, n)$-category of smooth, framed $n$-bordisms, $(n-1)$-dimensional boundaries, and corners down to points, is freely generated symmetric monoidal with ...
        6
        votes
        0answers
        182 views

        Differential topology on arbitrary fields

        What do the differential topology theories on arbitrary fields have in common? Different differential topology theories There is "ordinary" differential topology on real manifolds, with its rich ...
        5
        votes
        1answer
        278 views

        Is the connected sum of a triangulable manifold with a non-triangulable manifold a non-triangulable manifold?

        Let $M,N$ be topological manifolds such that $M$ does not admit a $PL$ structure and $N$ does. Is $M\#N$ still a triangulable manifold?
        7
        votes
        0answers
        207 views

        Atiyah duality with coefficients and boundary

        Looking at Atiyah's paper "Thom complexes", I find two statements of the Atiyah duality theorem: Proposition 3.2 Let $X$ be a compact smooth manifold with boundary $Y$ and tangent bundle $\tau$. Then ...
        19
        votes
        2answers
        1k views

        A manifold is a homotopy type and _what_ extra structure?

        Motivation: Surfaces Closed oriented 2-manifolds (surfaces) are "classified by their homotopy type". By this we mean that two closed oriented surfaces are diffeomorphic iff they're homotopy ...

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