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        Stack Exchange Network

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        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,...

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        30 views

        Bound on critical points of Lefschetz fibration with prescribed monodromy

        Let $\phi$ be a right-veering diffeomorphism of a surface $\Sigma$ of genus $g$ and $r$ boundary components. Suppose that the diffeomorphism is freely periodic so if $M$ is the associated open book ...
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        81 views

        Holomorphic map, Instantons of Complex Projective Space and Loop Group

        It seems that holomorphic (or rational) maps play a crucial role to relate the following data: Instanton in 1-dimensional complex Projective Space $$\mathbb{P}^1$$ in a 2 dimensional (2d) spacetime. ...
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        82 views

        Relating bordism generators in d and d+2 dimensions — an explicit example

        This is an attempt to make my relation between bordism invariants in $d$ and $d+2$ dimensions, following a previous attempt more explicit. This counts as a different question, since some more specific ...
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        1answer
        97 views

        Extending the maps between the bordism groups: NONE exsistence of a certain kind of extended group

        Let $M^d$ be a nontrivial bordism generator for the bordism group $$ \Omega_d^G= \mathbb{Z}_n, $$ suppose $G$ (like O, SO, Spin, etc) specify the group structure of the boridsm group. The $\mathbb{Z}...
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        125 views

        “Gerbes” in the cobordism theory

        In a lecture I attended today, I heard the use of gerbes in the cobordism theory. Previously, I use cobordism theory, but I never encounter the term "gerbes" when I work on bordism or cobordism group ...
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        62 views

        Relating bordism invairants in $d$ and $d+2$ dimensions

        Are there some relationship between mapping the bordism invairants of eq.1 and eq.2? $$\Omega_{O}^{d}(B(PSU(2^n)\rtimes\mathbb{Z}_2)) \tag{eq.1}$$ and $$\Omega_{O}^{d+2}(K(\mathbb{Z}/{2^n},2)) \...
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        1answer
        139 views

        Framings for 2-surgeries on 4-manifolds

        I'm interested in doing $2$-surgeries to $\sharp^k S^1 \times S^3$. That is to the manifold obtained from applying $1$-surgeries to $S^4$. Since $\pi_1(O(3)) = \mathbb{Z}_2$, there are two possible ...
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        1answer
        172 views

        Manifold generators of O-bordism invariants

        If I understand correctly, I can obtain the $O$-cobordism group of $$ \Omega^{O}_3(BO(3))=(\mathbb{Z}/2\mathbb{Z})^4, $$ The 3d cobordism invariants have 4 generators of mod 2 classes, are generated ...
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        Defining a notion of “volume of its lattice” for non-rational subspaces

        Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”: $$\...
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        1answer
        102 views

        Example of nonvanishing Waldhausen Nil group

        In a remarkable series of papers, both anticipating development in geometric topology and algebraic K-theory, specifically what we call now the Farrell-Jones conjecture, Waldhausen ...
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        2answers
        222 views

        Is every element of $Mod(S_{g,1})$ a composition of right handed Dehn twists?

        Let $S_{g,1}$ be the surface of genus $g \geq 1$ and $1$ boundary component. Let $Mod(S_{g,1})$ be the mapping class group in which we allow isotopies to rotate the action on the boundary (...
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        429 views

        Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$

        We fix $G=\mathrm{SL}_3(\mathbf{R})$. Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$? Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
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        52 views

        Topological Shape Operator More Sensitive than Inverse Limits

        This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...
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        73 views

        Simple invariants to detect concordance in general 3-manifolds

        Let $Y$ be a closed, connected, orientable 3-manifold. We call to oriented knots $K_1, K_2$ in $Y$ (smoothly) concordant if there is a smoothly, properly embedded annulus in $Y \times I$ such that ...
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        0answers
        100 views

        Weinstein neighborhood theorem for Lagrangians with Legendrian boundary

        $\require{AMScd}$ Weinstein's neighborhood theorem says that every Lagrangian has a standard neighborhood. The more precise statement goes like this. Theorem 1: (Lagrangian Neighborhood Theorem) ...

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