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        Questions tagged [gt.geometric-topology]

        Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

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        215 views

        “a result of Atiyah implies that the top cell of an orientable manifold splits off stably”

        I am looking for a reference for the following sentence “a result of Atiyah implies that the top cell of an orientable manifold splits off stably” Thank you very much for helping me find a reference ...
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        138 views

        Cohomology of $\operatorname{SO}(p,q;\mathbb{Z})$ with $p=3,q=19$

        I would like to understand the topology of the moduli space of Einstein orbifold metrics on the $K3$-surface. It is known that this space is given by the bi-quotient $SO(3,19;\mathbb{Z})\setminus SO(3,...
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        2answers
        827 views

        What is the generator of $\pi_9(S^2)$?

        This is more or less the same question as [ What is the generator of $\pi_9(S^3)$? ], except what I would like to know is if it is possible to describe this map in a way not only topologists can ...
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        votes
        1answer
        154 views

        Casson invariant and Euler characteristic

        A slogan I frequently hear is: "the Casson invariant is the Euler characteristic of the Floer homology of flat SU(2)-connections on the integral homology sphere". Is there a single paper/reference ...
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        votes
        1answer
        330 views

        Example of two exotic closed 4-manifolds s.t. SW(X)=0

        I am interested in seeing examples of two closed 4-manifolds $X_1,X_2$ such that $SW(X_i)=0$ and they are homeomorphic but not diffeomorphic. So far in the literature I've only found examples which ...
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        0answers
        50 views

        Order of separating Dehn twists in the image of Johnson homomorphisms

        Let $S$ be a closed surface, $\Gamma=\pi_1 S$ and $\Gamma_i$ be the lower central series defined by $\Gamma_0=\Gamma$ and $\Gamma_i=[\Gamma,\Gamma_{i-1}].$ The Johnson filtration $\lbrace \mathcal{K}...
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        1answer
        219 views

        History of the notion of $(G,X)$-structure

        I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work. So far, it appears that he was the first to set it. Many mathematicans ...
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        60 views

        Asking SnapPy for core curves after surgery

        Suppose I give SnapPy a cusped hyperbolic 3-manifold (using, say, the link editor) and specify some filling. SnapPy can then provide a presentation of the fundamental group of the filled manifold. Can ...
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        81 views

        “Brunnian” words in solvable groups

        Let $G$ be a group, and call a word $W(x_1,\dots,x_n)$ in letters $x_i$ and $x_i^{-1}$ "$G$-Brunnian" if there exist $g_1,\dots,g_n\in G$ with $W(g_1,\dots,g_n)\neq1$, but $W(h_1,\dots,h_n)=1$ as soon ...
        10
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        1answer
        81 views

        What are the possible linking matrices of a quasi-positive link?

        I was surprised recently to come across a 3-component link where the linking number of two of the components was negative. For a while I thought I had made a mistake, then I thought a little more and ...
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        0answers
        69 views

        source of argument about relative primeness of simple closed curves on tori

        I have know this argument for decades. I have no idea of its source. If anyone knows (not guesses) its origin, then I would be very appreciative. My guesses are among Ralph Fox, JHC Whitehead, RH ...
        3
        votes
        0answers
        91 views

        The bridge index and crookedness of a knot

        I am reading Dale Rolfsen's book KNOTS AND LINKS, at page 115, I can't figure out why the crookedness of a knot equals its bridge index. Please give me some hints or any references available, much ...
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        0answers
        117 views

        Stabilizing an open book with Anosov piece

        It was proven by Colin and Honda in Stabilizing the monodromy of an open book decomposition that any diffeomorphism can be made pseudo-Anosov and right-veering after a series of positive ...
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        votes
        0answers
        94 views

        Are triangulations with common refinements PL-homeomorphic?

        Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...
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        votes
        0answers
        142 views

        smooth structure on complete intersection

        A complete intersection is an algebraic variety cut out by homogenous polynomials. Geometrically, this is the intersection of hypersurfaces in complex projective space. Below, let's confine to the ...

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