Active questions tagged gt.geometric-topology - MathOverflowmost recent 30 from www.4124039.com2019-04-17T18:27:54Zhttp://www.4124039.com/feeds/tag?tagnames=gt.geometric-topology&sort=newesthttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://www.4124039.com/q/3283064Wu formula for manifolds with boundaryBorromeanhttp://www.4124039.com/users/1025152019-04-17T17:05:28Z2019-04-17T17:05:28Z
<p>The classical Wu formula claims that if <span class="math-container">$M$</span> is a smooth closed <span class="math-container">$n$</span>-manifold with fundamental class <span class="math-container">$z\in H_n(M;\mathbb{Z}_2)$</span>, then the total Stiefel-Whitney class <span class="math-container">$w(M)$</span> is equal to <span class="math-container">$Sq(v)$</span>, where <span class="math-container">$v=\sum v_i\in H^*(M;\mathbb{Z}_2)$</span> is the unique cohomology class such that
<span class="math-container">$$\langle v\cup x,z\rangle=\langle Sq(x),z\rangle$$</span>
for all <span class="math-container">$x\in H^*(M;\mathbb{Z}_2)$</span>. Thus, for <span class="math-container">$k\ge0$</span>, <span class="math-container">$v_k\cup x=Sq^k(x)$</span> for all <span class="math-container">$x\in H^{n-k}(M;\mathbb{Z}_2)$</span>, and
<span class="math-container">$$w_k(M)=\sum_{i+j=k}Sq^i(v_j).$$</span>
Here the Poincare duality guarantees the existence and uniqueness of <span class="math-container">$v$</span>.</p>
<p>My question: if <span class="math-container">$M$</span> is a smooth compact <span class="math-container">$n$</span>-manifold with boundary, is there a similar Wu formula?
In this case, there is a fundamental class <span class="math-container">$z\in H_n(M,\partial M;\mathbb{Z}_2)$</span> and the relative Poincare duality claims that capping with <span class="math-container">$z$</span> yields duality isomorphisms
<span class="math-container">$$D:H^p(M,\partial M;\mathbb{Z}_2)\to H_{n-p}(M;\mathbb{Z}_2)$$</span>
and
<span class="math-container">$$D:H^p(M;\mathbb{Z}_2)\to H_{n-p}(M,\partial M;\mathbb{Z}_2).$$</span></p>
<p>Thank you!</p>
http://www.4124039.com/q/3280163Covariant derivative of determinant of the metric tensorPhilliphttp://www.4124039.com/users/682792019-04-14T09:51:47Z2019-04-17T14:16:28Z
<p>Let <span class="math-container">$(M,g)$</span> be a Riemannian manifold and <span class="math-container">$g$</span> the Riemannian metric in coordinates <span class="math-container">$g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$</span>, where <span class="math-container">$x^{i}$</span> are local coordinates on <span class="math-container">$M$</span>. Denote by <span class="math-container">$g^{\alpha \beta}$</span> the inverse components of the inverse metric <span class="math-container">$g^{-1}$</span>. Let <span class="math-container">$\nabla$</span> be the Levi-Civita connection of the metric <span class="math-container">$g$</span>. Consider, locally, the function <span class="math-container">$\det((g_{\alpha \beta})_{\alpha \beta})$</span>. It is known that <span class="math-container">$\nabla \det((g_{\alpha \beta})_{\alpha \beta}) = 0$</span> by using normal coordinates etc...</p>
<p>I would like to show this fact without using normal coordinates. Just by computation. Here is what I have so far: </p>
<p><span class="math-container">$\nabla \det((g_{\alpha \beta})_{\alpha \beta}) = \left [ g^{\gamma \delta} \partial_{\delta} \det((g_{\alpha \beta})_{\alpha \beta}) \right ] \partial_{\gamma} = \left [ \det((g_{\alpha \beta})_{\alpha \beta}) g^{\gamma \delta} g^{\beta \alpha} \partial_{\delta} g_{\alpha \beta}\right ] \partial_{\gamma}.$</span>`</p>
<p>Here: the first equality sign follows from the definition of the gradient of a function and the second equality sign is the derivative of the determinant.</p>
<p><strong>Question:</strong> How do I continue from here without using normal coordinates? Or are there any mistakes? If yes, where and which?</p>
<p>Greetings,
Phil</p>
http://www.4124039.com/q/3282091Is there a method to cut a hypercube into disjoint cubes [on hold]Dreamer123http://www.4124039.com/users/1379032019-04-16T17:21:23Z2019-04-17T01:28:27Z
<p>Since Borsuk conjecture hold for centrally symmetric convex sets in <span class="math-container">$\mathbb{R}^n$</span>
so we can cut a hypercube into at least <span class="math-container">$n+1$</span> disjoint parts.</p>
<p>Is there a method how can one do that?</p>
http://www.4124039.com/q/3281290Convergence of slices in some topology on hyperspace of closed setsIian Smythehttp://www.4124039.com/users/161072019-04-15T15:25:00Z2019-04-15T15:44:39Z
<p>Let <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> be metrizable spaces, <span class="math-container">$Y$</span> compact, and <span class="math-container">$C\subseteq X\times Y$</span> closed. </p>
<p>For each <span class="math-container">$y\in Y$</span>, let <span class="math-container">$C^y=\{x\in X:(x,y)\in C\}$</span>, the <em><span class="math-container">$y$</span>-slice</em> of <span class="math-container">$C$</span>. Since <span class="math-container">$Y$</span> is compact, the projection from <span class="math-container">$X\times Y$</span> onto <span class="math-container">$X$</span> is a closed map, and it follows that each <span class="math-container">$C^y$</span> is closed in <span class="math-container">$X$</span>.</p>
<p>Here is my question: Suppose that <span class="math-container">$(y_n)$</span> is a sequence in <span class="math-container">$Y$</span> converging to some <span class="math-container">$y\in Y$</span>. Is there a "reasonable" topology on the hyperspace of closed subsets of <span class="math-container">$X$</span> such that <span class="math-container">$(C^{y_n})$</span> converges to <span class="math-container">$C^y$</span>?</p>
<p>"Reasonable" is up to interpretation here, but I would like it to be metrizable and closely related to the topology on <span class="math-container">$X$</span>.</p>
<p>If it helps, you can take <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> to be complete separable metric spaces and <span class="math-container">$C$</span> clopen. However, for the example I have in mind, <span class="math-container">$X$</span> is very much not compact (e.g., <span class="math-container">$X=\mathbb{N}^\mathbb{N}$</span>).</p>
http://www.4124039.com/q/3280620Cutting a smooth convex body into $n$ bodies [on hold]Dreamer123http://www.4124039.com/users/1379032019-04-14T19:14:08Z2019-04-14T20:17:54Z
<p>If we have an <span class="math-container">$n$</span>-dimensional body which is both convex and smooth. If we cut it down to <span class="math-container">$n$</span> parts. Is there any result in the literature that guarantees that these <span class="math-container">$n$</span> parts are also convex and smooth?</p>
<p>Thanks</p>
http://www.4124039.com/q/3278128open book decompositions of $\Sigma\times S^1$Mark Granthttp://www.4124039.com/users/81032019-04-11T19:50:43Z2019-04-14T06:31:55Z
<p>Let <span class="math-container">$\Sigma$</span> be a <em>closed</em> orientable surface. Is there a standard open book decomposition on the <span class="math-container">$3$</span>-manifold <span class="math-container">$M=\Sigma\times S^1$</span>?</p>
<p>If the binding is allowed to be empty in the definition of an open book decomposition, then this is obvious, since <span class="math-container">$M$</span> is the mapping torus of the identity of <span class="math-container">$\Sigma$</span>. The literature is not clear on this, however.</p>
<p>If the binding must be non-empty, then since it is contained in every page, it seems that the obvious fibering <span class="math-container">$M\to S^1$</span> is never the fibering of an open book. </p>
http://www.4124039.com/q/3247564Hyperbolic Dehn surgeries and SU(2)-representationsThiKuhttp://www.4124039.com/users/390822019-03-06T08:48:16Z2019-04-14T06:12:30Z
<p>Let <span class="math-container">$S^3-K$</span> be the complement of the figure eight knot complement. Thurston, in his Lecture Notes, constructed a hyperbolic structure, which comes from a discrete, faithful representation <span class="math-container">$\pi_1(S^3-K)\to SL(2,{\mathcal O}_3)\subset SL(2,C)$</span>. This representation is not conjugate into <span class="math-container">$SU(2)$</span>, because traces are not real. But, of course, one can get nontrivial <span class="math-container">$SU(2)$</span>-representations factoring over the abelianization <span class="math-container">$Z$</span> of <span class="math-container">$\pi_1(S^3-K)$</span>.</p>
<p>Thurston goes on to prove that almost all Dehn surgeries at the figure eight knot complement admit a hyperbolic structure, hence a faithful, discrete representation <span class="math-container">$\pi_1(S^3-K)\to PSL(2,C)$</span>, which by Culler lifts to <span class="math-container">$SL(2,C)$</span> and again is not conjugate into <span class="math-container">$SU(2)$</span>.</p>
<p>On the other hand, Kronheimer and Mrowka show that for Dehn surgery coefficients <span class="math-container">$\vert\frac{p}{q}\vert\le 2$</span>, the Dehn surgered manifold admits a noncyclic representation to <span class="math-container">$SU(2)$</span>. This is, in a sense, complementary to Thurston¡®s result, who proved his result for sufficiently large Dehn surgery coefficients.</p>
<p>Question: is it known whether Dehn surgeries at the figure eight knot complement, with sufficiently large Dehn surgery coefficients, admit nontrivial (not necessarily faithful) representations to <span class="math-container">$SU(2)$</span>?</p>
http://www.4124039.com/q/3278003Rational homology cobordism invariantsO?uz ?avkhttp://www.4124039.com/users/1311722019-04-11T17:47:55Z2019-04-13T19:24:46Z
<p>As far as I searched, I couldn't find, but I wanna ask that:</p>
<p>Is there any rational homology cobordism invariant different than Ozsváth-Szabó <span class="math-container">$d$</span>-invariant and Frøyshov <span class="math-container">$h$</span>-invariant?</p>
<p><strong>Edit:</strong></p>
<p>A closed oriented <span class="math-container">$3$</span>-manifold <span class="math-container">$Y$</span> is called <strong>rational homology sphere</strong> if <span class="math-container">$H_{*}(Y,\mathbb{Q})= H_{*}(S^{3},\mathbb{Q})$</span>. Let <span class="math-container">$Y_1$</span> and <span class="math-container">$Y_2$</span> be rational homology spheres. They are said to be <strong>homology cobordant</strong>, if there exits a smooth compact oriented <span class="math-container">$4$</span>-manifold <span class="math-container">$X$</span> with boundary <span class="math-container">$\partial X= (-Y_1) \cup Y_2$</span> such that <span class="math-container">$H_*(X,Y_i; \mathbb Q)=0$</span> for <span class="math-container">$i=0,1$</span>.</p>
<p>Then a <strong>rational homology cobordism invariant</strong> is an invariant of rational homology spheres which does not change under homology cobordism. </p>
http://www.4124039.com/q/3273386Milnor immersion of circle, disks, and a ballChris Gerighttp://www.4124039.com/users/123102019-04-06T17:32:06Z2019-04-13T19:05:21Z
<p>Milnor surprisingly found an immersion of a circle in the plane which bounds two "incompatible" immersed disks (see [<a href="https://books.google.com/books?id=Jyj7CAAAQBAJ&pg=PA28&lpg=PA28&dq=milnor+immersion+circle+disks&source=bl&ots=BsDZoA7Rkx&sig=ACfU3U2MG6TtFsOccPuXInjgi5ytk098CQ&hl=en&sa=X&ved=2ahUKEwjR1MLC_bvhAhWKTd8KHdr6B4oQ6AEwBHoECAcQAQ#v=onepage&q=milnor%20immersion%20circle%20disks&f=false" rel="nofollow noreferrer">1</a>] for example, picture included). I think I can glue these two disks together to define an immersion of a sphere <span class="math-container">$f:S^2\to\mathbb{R}^3$</span> where Milnor's immersed circle (the equator) sits in the <span class="math-container">$xy$</span>-plane and contracts to a point (in the <span class="math-container">$\pm z$</span> directions) by running along either immersed disk.</p>
<p><strong>Does <span class="math-container">$f(S^2)$</span> bound an immersed ball, i.e. does <span class="math-container">$f$</span> extend to an immersion of a 3-ball?</strong></p>
<p>This question was sparked by glancing at Gromov's book [<a href="https://books.google.com/books?id=Jyj7CAAAQBAJ&pg=PA28&lpg=PA28&dq=milnor+immersion+circle+disks&source=bl&ots=BsDZoA7Rkx&sig=ACfU3U2MG6TtFsOccPuXInjgi5ytk098CQ&hl=en&sa=X&ved=2ahUKEwjR1MLC_bvhAhWKTd8KHdr6B4oQ6AEwBHoECAcQAQ#v=onepage&q=milnor%20immersion%20circle%20disks&f=false" rel="nofollow noreferrer">1</a>] and a related paper of Eliashberg-Mishachev which argues that the projection <span class="math-container">$S^2\to\mathbb{R}^2$</span> (composing <span class="math-container">$f$</span> above with projection onto <span class="math-container">$xy$</span>-plane) is not homotopic through folded maps to the "standard" folded map <span class="math-container">$(x,y,z)\mapsto (x,y)$</span> on the unit 2-sphere (the fold being the equator).</p>
http://www.4124039.com/q/3152800Iteration of a parabolic isometry [on hold]AVATARhttp://www.4124039.com/users/902982018-11-14T07:05:19Z2019-04-13T10:08:10Z
<p>Consider the real hyperbolic <span class="math-container">$n$</span>-space in its conformal ball model.</p>
<p>Let <span class="math-container">$a$</span> be a point of the boundary <span class="math-container">$S^{n-1}$</span> fixed by a parabolic transformation (an isometry which fixes exactly one point in the sphere and no point in the open ball) <span class="math-container">$\phi$</span> of <span class="math-container">$B^{n}$</span>. One has to show that if <span class="math-container">$x$</span> is in <span class="math-container">$\bar{B^{n}}$</span> , then </p>
<p><span class="math-container">$$\lim_{m \rightarrow \infty} \phi^{m}(x)=a$$</span></p>
<p>I have tried taking the convergent subsequence using compactness of <span class="math-container">$\bar{B^{n}}$</span> but could not prove the required. Any kind of help would be appreciated. </p>
<p>P.S. : I posted this question few days back on MathStack but didn't get any satisfactory answer. </p>
http://www.4124039.com/q/32790911Are "most" spaces aspherical?Tim Campionhttp://www.4124039.com/users/23622019-04-12T17:17:11Z2019-04-13T03:34:40Z
<p>There's a heuristic idea that "most" closed manifolds <span class="math-container">$M$</span> are aspherical (i.e. <span class="math-container">$\pi_{\geq 2}(M) = 0$</span>). Does this heuristic extend usefully to all spaces -- or at least to all finite CW complexes?</p>
<p>To make this question more precise, I should say something about in what sense "most" manifolds are aspherical. I don't know a lot about this heuristic, but here's where I'm coming from:</p>
<ul>
<li><p>It's true in low dimensions: trivially in 0 or 1 dimensions, and by classification of surfaces in 2 dimensions. In 3 dimensions, I've heard it said that part of the upshot of Thurston's Geometrization Conjecture is that "most" 3-manifolds are hyperbolic, and in particular aspherical.</p></li>
<li><p>There's some discussion of this heuristic in <a href="https://www.him.uni-bonn.de/lueck/data/lueck_survey_on_aspherical_manifolds_xxx0902.2480v3.pdf" rel="noreferrer">this survey article of Luck</a> (at the end).</p></li>
</ul>
<p>How do things look if we think about CW complexes? Well, every 0 or 1-dimensional CW complex is aspherical. And the Kan-Thurston theorem tells us that every space is homology-equivalent to an aspherical space. But it's really not clear to me whether I should think of "most" spaces as being aspherical.</p>
http://www.4124039.com/q/3279172Existence of smooth structures on topological $3$-manifolds with boundaryDennishttp://www.4124039.com/users/1382992019-04-12T18:30:02Z2019-04-12T18:30:02Z
<p>It is said in this thread <a href="http://www.4124039.com/questions/296171/unique-smooth-structure-on-3-manifolds?rq=1">Unique smooth structure on <span class="math-container">$3$</span>-manifolds</a> that every topological <span class="math-container">$3$</span>-manifold admits a smooth structure. However it is not specified whether the manifolds are allowed to have boundary or not and the references therein are hard to digest (since I am no expert in this field). So my question is whether every topological <span class="math-container">$3$</span>-manifold with boundary admits a smooth structure?</p>
<p>By smooth structure I mean a collection of charts mapping homeomorphically onto open subsets of the (closed) upper half space, whose transition functions extend to smooth functions on some open neighbourhood of their domains.</p>
<p>I believe that the answer to my question is yes (assuming the theorem holds for topological <span class="math-container">$3$</span>-manifolds without boundary) with the following reasoning:</p>
<p>Let <span class="math-container">$M$</span> be a topological <span class="math-container">$3$</span>-manifold with non-empty boundary. Then we can consider the boundaryless double <span class="math-container">$\widetilde{M}$</span> of <span class="math-container">$M$</span>, which is a topological <span class="math-container">$3$</span>-manifold without boundary in which <span class="math-container">$M$</span> can be (topologically) embedded. We can now equip <span class="math-container">$\widetilde{M}$</span> with a smooth structure and restrict the charts to the embedding of <span class="math-container">$M$</span>, which gives us a smooth atlas of the embedding and in turn a smooth atlas of <span class="math-container">$M$</span>.</p>
<p>Is my reasoning correct?</p>
<p>Thanks a lot in advance!</p>
http://www.4124039.com/q/3277167The (co)tangent sheaf of a topological spaceQfwfqhttp://www.4124039.com/users/47212019-04-10T20:24:14Z2019-04-11T21:11:23Z
<p>Let <span class="math-container">$X$</span> be a topological space (assume additional assumptions if needed) and denote by <span class="math-container">$\mathcal O _X$</span> its sheaf of <span class="math-container">$\Bbbk$</span>-valued continuous functions where <span class="math-container">$\Bbbk$</span> is <span class="math-container">$\mathbb{R}$</span> or <span class="math-container">$\mathbb{C}$</span> with standard topology.</p>
<p>Then, as it is done in the differentiable setting or in algebraic geometry, one can define the following objects
<span class="math-container">$$T_X:=\mathscr{Der}_\Bbbk (\mathcal O_X,\mathcal O_X)$$</span>
the tangent sheaf, i.e. the sheaf of <span class="math-container">$\Bbbk$</span>-linear derivations of <span class="math-container">$\mathcal O_X$</span> with values in <span class="math-container">$\mathcal O_X$</span> (on local sections, <span class="math-container">$\Bbbk$</span>-linear maps <span class="math-container">$D:\mathcal O_X(U)\to\mathcal O_X(U)$</span> satisfying Leibniz: <span class="math-container">$D(f\cdot g)=f\cdot Dg + g\cdot Df$</span>), and
<span class="math-container">$$\Omega_X^1:=\mathcal I/\mathcal I^2$$</span></p>
<p>the sheaf of differentials, where <span class="math-container">$\mathcal I$</span> is the ideal sheaf of <span class="math-container">$X$</span> embedded diagonally <span class="math-container">$\Delta:X\hookrightarrow X\times X$</span> into <span class="math-container">$X\times X$</span> (i.e. <span class="math-container">$\mathcal I(U)=$</span> functions in <span class="math-container">$\mathcal O_{X\times X}(U)$</span> that are zero on every point of <span class="math-container">$\Delta(X)\subset X\times X$</span>).</p>
<p>Well, what can be said about these two sheaves? Anything interesting at all?</p>
<p>Also, is there any relationship between <span class="math-container">$T_X$</span> and the "tangent microbundle" <span class="math-container">$\tau_X$</span> in case <span class="math-container">$X$</span> is a topological manifold?</p>
http://www.4124039.com/q/22067612References for Stiefel-Whitney class of Stiefel manifolds and GrassmanniansQSRhttp://www.4124039.com/users/658002015-10-12T01:06:24Z2019-04-11T18:35:43Z
<p>Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$
$$
w(M)=1+w_1(TM)+w_2(TM)+\cdots
$$
I want to find references for
$$
w(SO(n)/SO(k)),(i.e. w(V_{n-k}(\mathbb{R}^n))), \\
w(U(n)/U(k)),(i.e. w(V_{n-k}(\mathbb{C}^n))), \\
w(Sp(n)/Sp(k)),(i.e. w(V_{n-k}(\mathbb{H}^n))),\\
w(SO(n)/(SO(k)\times SO(n-k))),(i.e. w(G_{n-k}(\mathbb{R}^n))), \\
w(U(n)/(U(k)\times U(n-k))),(i.e. w(G_{n-k}(\mathbb{C}^n))), \\
w(Sp(n)/(Sp(k)\times Sp(n-k))),(i.e. w(G_{n-k}(\mathbb{H}^n))).
$$
Which ones of the above are known? Where could I find these formulas?</p>
http://www.4124039.com/q/1041268Knot theory without planar diagrams?Marius Buligahttp://www.4124039.com/users/77722012-08-06T17:53:37Z2019-04-11T17:00:52Z
<p>I remember reading somewhere that there are just a few contributions to knot theory which do not involve knot diagrams. Hence my question: </p>
<p>Does anybody know about papers concerning knot theory which work directly with knots in 3D, without using planar (projections, for example) diagrams? </p>
http://www.4124039.com/q/3276515"Dimension" of discrete subgroups of infinite covolume in Lie groupsYCorhttp://www.4124039.com/users/140942019-04-10T08:22:23Z2019-04-11T15:22:13Z
<p>Let <span class="math-container">$G$</span> be a semisimple Lie group with finite center, <span class="math-container">$K$</span> a maximal compact
subgroup, and <span class="math-container">$d=\dim(G/K)$</span>. Let <span class="math-container">$\Gamma$</span> be a non-cocompact discrete subgroup of <span class="math-container">$G$</span>. [Edit: assume that <span class="math-container">$\Gamma$</span> is virtually torsion-free (which is automatic if <span class="math-container">$G$</span> is linear and <span class="math-container">$\Gamma$</span> is finitely generated).]</p>
<blockquote>
<p>Is it true that the virtual cohomological dimension (vcd) of <span class="math-container">$\Gamma$</span> is
<span class="math-container">$<d$</span>?</p>
</blockquote>
<p>The virtual cohomological dimension in this case is the cohomological dimension of some/every torsion-free finite-index subgroup. I guess it's also the rational cohomological dimension.</p>
<p>Remarks:</p>
<ul>
<li>if <span class="math-container">$\Gamma$</span> is cocompact the vcd equals <span class="math-container">$d$</span>;</li>
<li>in general, the vcd is <span class="math-container">$\le d$</span>;</li>
<li>if <span class="math-container">$\Gamma$</span> is a non-cocompact lattice, then the vcd is <span class="math-container">$<d$</span> (at least in
the arithmetic case, where it's related to the <span class="math-container">$\mathbf{Q}$</span>-rank, cf work of
Borel-Serre, and also in the rank-1 case; I think the general case follows).</li>
</ul>
<p>I'd also be interested by variants of this question, where vcd is replaced
by the asymptotic dimension, or by Roe's coarse cohomological dimension.</p>
http://www.4124039.com/q/10445114Irreducible homology 3-spheres that bound smooth contractible manifoldsIgor Belegradekhttp://www.4124039.com/users/15732012-08-11T00:24:51Z2019-04-10T09:09:45Z
<p>Some examples of irreducible homology 3-spheres that bound <b> smooth</b> contractible 4-manifolds are listed in the comment to problem 4.2 in <a href="http://math.berkeley.edu/~kirby/problems.ps.gz"> Kirby's problem list</a>, and all of them happen to occur among the <a href="http://www.maths.ed.ac.uk/~aar/papers/milnbries.pdf">
Brieskorn spheres</a> $\Sigma(p,q,r)$ modelled on $\widetilde{SL}_2(\mathbb R)$,
i.e. such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$ (except for the standard $S^3$, of course). </p>
<p><b> Question.</b> Are there any known homology 3-spheres that bound smooth contractible 4-manifolds and are modelled on other geometries, e.g. NIL or the hyperbolic space?</p>
<p>EDIT: Glazner in the paper "Uncountably many contractible 4-manifolds" constructed some other examples but I cannot recognize the geometry. (Glazner's six page paper
is easily googlable by title, and it gives an explicit representation for the fundamental group, denoted $G_n$ on page 40). </p>
http://www.4124039.com/q/3275148Number of Reflections in a Circle between Two PointsArgonhttp://www.4124039.com/users/943982019-04-08T20:03:18Z2019-04-10T00:05:29Z
<p>For my research I am interested in the transmission characteristics between a transmitter (Tx) and a receiver (Rx) situated in a circular room. In particular, it is important for me to know the number of paths a ray can take such that it reflects <em>exactly once</em> off the walls of the room. </p>
<p><a href="https://i.stack.imgur.com/xmJ7E.png" rel="noreferrer"><img src="https://i.stack.imgur.com/xmJ7E.png" alt="Reflections from a transmitter to a receiver in a room"></a></p>
<p>Since reflections occur such that the incident ray has the same angle relative to the normal as the reflected ray, I tried to use vectors to attack the problem but the math became very unwieldy.</p>
<p>Empirically, I have found that depending on the situation of the transmitter and receiver, there could be 2, 3, or 4 paths¡ªno more, no less. There is an exceptional case where the transmitter and receiver are co-located at the centre, in which case there are infinitely many paths.</p>
<p>Can my experimental result be validated (or denied) analytically?</p>
http://www.4124039.com/q/3274643Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$Lucas Kaufmannhttp://www.4124039.com/users/354282019-04-08T08:27:35Z2019-04-08T22:24:05Z
<p>Let <span class="math-container">$R$</span> be a subset of <span class="math-container">$\text{PSL}_2(\mathbb C)$</span> and consider its natural action on <span class="math-container">$\mathbb {CP}^1$</span>. We say that <span class="math-container">$R$</span> is elementary if either <span class="math-container">$R$</span> is conjugated to a subset of <span class="math-container">$\text{SU(2)}$</span> or if there is a finite subset of <span class="math-container">$\mathbb {CP}^1$</span> which is invariant by every element of <span class="math-container">$R$</span>.</p>
<p>For <span class="math-container">$n \geq 1$</span> set <span class="math-container">$R^n : = \{g_1\cdots g_n : g_i \in R\}$</span> (that is, the set of products of elements in <span class="math-container">$R$</span> of length <span class="math-container">$n$</span>) and <span class="math-container">$R^n \cdot R^{-n}:=\{gh^{-1} : g,h \in R^n\}$</span>. I'm looking for a reference for the following statement, which seems to be true:</p>
<p><em>Assume <span class="math-container">$R$</span> is non-elementary. Then there exists an <span class="math-container">$N \geq 1$</span> and an <span class="math-container">$M \geq1$</span>, such that <span class="math-container">$R^{N}$</span> contains a loxodromic element and <span class="math-container">$R^{M} \cdot R^{-M}$</span> contains a non-elliptic element.</em></p>
<p>I have a somewhat tedious proof of this fact but I would guess that such a result is well-known (at least the one about <span class="math-container">$R^N$</span>). A reference to a book or a paper containing these results or closely related ones would be of great help.</p>
<p>Thanks in advance for your answers.</p>
http://www.4124039.com/q/32737210Mapping classes as Lefschetz fibrations over surfaces with positive genusPaulhttp://www.4124039.com/users/430972019-04-07T01:58:52Z2019-04-07T01:58:52Z
<p>Let <span class="math-container">$\Sigma_{g,r}$</span> be the surface of genus <span class="math-container">$g$</span> and <span class="math-container">$r$</span> boundary components. It is known that, from a positive factorization of a mapping class <span class="math-container">$\phi$</span> in the mapping class group <span class="math-container">$MCG(\Sigma_{g,r}, \partial \Sigma_{g,r})$</span>, one can construct a Lefschetz fibration (with corners) over the disc <span class="math-container">$D^2$</span> so that there are as many critical points/values as Dehn twists take place in the factorization and such that the restriction of the Lefschetz fibration to <span class="math-container">$\partial D^2$</span> is a surface bundle with monodromy <span class="math-container">$\phi$</span>. Conversely, from any such fibration one can recover a mapping class factorized by right-handed Dehn twists.</p>
<p>My question is:</p>
<blockquote>
<p>Is it known for which mapping classes <span class="math-container">$\phi \in MCG(\Sigma_{g,r}, \partial \Sigma_{g,r})$</span> there exists <span class="math-container">$h \in \mathbb{N}$</span> and a Lefschetz fibration over the surface <span class="math-container">$\Sigma_{h,1}$</span> such that the monodromy of the restriction of the Lefschetz fibration to <span class="math-container">$\partial \Sigma_{h,1}$</span> is <span class="math-container">$\phi$</span>?</p>
</blockquote>
http://www.4124039.com/q/3271712A Backtrack as a Single Word in a Group Presentation yields a Complex that isn't of the Same Homotopy Type?lunchmeathttp://www.4124039.com/users/1191812019-04-04T22:20:11Z2019-04-05T13:19:20Z
<p>By "backtrack" I mean a subword of a relator in a group presentation of the form <span class="math-container">$x x^{-1}$</span>. </p>
<p>Let <span class="math-container">$X = \langle a \rangle$</span> as a presentation complex. </p>
<p>Let <span class="math-container">$Y = \langle a$</span> | <span class="math-container">$aa^{-1} \rangle$</span> as a presentation complex. </p>
<p>Now we see that <span class="math-container">$X$</span> is a circle and <span class="math-container">$Y$</span> is a pinched torus, and these two spaces clearly do not have the same Homotopy Type as <span class="math-container">$\pi_2(Y)$</span> is nontrivial. </p>
<p>However it was said in "<a href="https://link.springer.com/article/10.1023%2FA%3A1019693615888" rel="nofollow noreferrer">A Covering Space With no Compact Core</a>" (<a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1934009" rel="nofollow noreferrer">MSN</a>) by Daniel Wise that:</p>
<p><span class="math-container">$\langle a, b, t $</span> | <span class="math-container">$ [a,b]^t = [a,b][b,a] \rangle$</span> is homotopy equivalent to <span class="math-container">$\langle a, b, t $</span> | <span class="math-container">$[a,b] \rangle$</span>. </p>
<p>Is this always true when the backtrack is a proper subword of a relator? Is the above case with <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> the only real nonexample? </p>
http://www.4124039.com/q/31749412An almost complex structure on $S^2\times ...\times S^2 / \mathbb{Z_2}$aglearnerhttp://www.4124039.com/users/134412018-12-12T12:55:01Z2019-04-04T22:12:20Z
<p>Consider the product of <span class="math-container">$2n$</span> two-spheres <span class="math-container">$X_n=(S^2)^{2n}$</span>. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) central symmetry for each <span class="math-container">$S^2$</span>. Is it possible to say for which <span class="math-container">$n$</span> the quotient manifold <span class="math-container">$X_n/\mathbb Z_2$</span> admits an almost complex structure? I believe that if such <span class="math-container">$n$</span> exists then <span class="math-container">$n>1$</span>, i.e. <span class="math-container">$S^2\times S^2/\mathbb Z_2$</span> is not almost complex (for the defined action).</p>
http://www.4124039.com/q/3271090Reference request on Borsuk conjecture [closed]Dreamer123http://www.4124039.com/users/1379032019-04-04T03:47:08Z2019-04-04T04:05:08Z
<p>I just heard of Borsuk conjecture. I want to ask if there are any references preferably looking at the problem from the point of view of Mathematical analysis I can study it from? </p>
<p>Thanks</p>
http://www.4124039.com/q/3262524Chirality and Anti-Chirality of links in 3 and in 5 dimensionswonderichhttp://www.4124039.com/users/270042019-03-25T01:56:34Z2019-04-03T16:04:57Z
<p>We know there is a chiral knot which is a knot that is not equivalent to its mirror image. It is well known in the mathematical field of knot theory:
<a href="https://en.wikipedia.org/wiki/Chiral_knot" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Chiral_knot</a></p>
<p><a href="https://i.stack.imgur.com/KA3rT.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/KA3rT.png" alt="enter image description here"></a></p>
<blockquote>
<p>My first question is about </p>
<p>(1) the literature and the References on </p>
<ul>
<li><p>the chirality of link in 3 dimensions</p></li>
<li><p>the chirality of link in 5 dimensions</p></li>
</ul>
<p>What are some good text/Refs on the topological invariants of these chiralities of links of 1-submanifolds in 3 dimensions? (addressed somewhere in the literature?) </p>
</blockquote>
<p>For example, in 5 dimensions, let me consider a 5-sphere <span class="math-container">$S^5$</span>. Let me define a new quartic link Q of 5-dimensions in <span class="math-container">$S^5$</span>: such that <span class="math-container">$\Sigma^3_{W_{{(i)}}},\Sigma^3_{W_{{(ii)}}},\Sigma^3_{W_{{(iii)}}}$</span> are 3 sets of 3-submanifolds, while the <span class="math-container">$\Sigma^2_U$</span> is a 2-surface. Let <span class="math-container">$V^4_{W_{{(i)}}}, V^4_{W_{{(ii)}}}, V^4_{W_{{(iii)}}}, V^3_U$</span> be their Seifert volumes in one higher dimensions.</p>
<blockquote>
<p>Are these chiralities of link of 2-submanifolds and 3-submanifolds in 5 dimensions also addressed somewhere in the literature? </p>
</blockquote>
<p>There can be a link invariant defined in this manner:
<span class="math-container">$$
{ \#(V^4_{W_{{(i)}}}\cap V^4_{W_{{(ii)}}}\cap V^4_{W_{{(iii)}}}\cap V^3_U)\equiv\text{Q}^{(5)}(\Sigma^3_{W_{{(i)}}},\Sigma^3_{W_{{(ii)}}},\Sigma^3_{W_{{(iii)}}},\Sigma^2_U)}
$$</span></p>
<p>I am suspecting there could be a opposite chirality of this invariant defined as <span class="math-container">$\overline{\text{Q}^{(5)}}$</span> as follows:
<span class="math-container">$$
{ \#(V^4_{W_{{(ii)}}}\cap V^4_{W_{{(i)}}}\cap V^4_{W_{{(iii)}}}\cap V^3_U)+\dots \equiv\overline{\text{Q}^{(5)}}(\Sigma^3_{W_{{(ii)}}},\Sigma^3_{W_{{(i)}}},\Sigma^3_{W_{{(iii)}}},\Sigma^2_U)}
$$</span></p>
<blockquote>
<p>(2) Do similar <strong>chirality and anti-chirality of link invariants</strong> in 3 dimensions, happening for example to Borromean rings? Or other Brunnian links? Examples and References are welcome. </p>
</blockquote>
http://www.4124039.com/q/3261853Triangulations of 3-manifolds in Regina and SnapPyIgor Rivinhttp://www.4124039.com/users/111422019-03-24T06:00:57Z2019-04-02T17:40:23Z
<p>I have been doing some statistical studies on small 3-manifolds, and I note that one can produce larg-ish censuses of triangulations in Regina. Now, the Regina documentation tells us how to convert a single triangulation into SnapPy format, but is mum on any batch way of doing this. Any help appreciated...</p>
http://www.4124039.com/q/8075810Embeddings without nonvanishing normal vector fieldsMike Usherhttp://www.4124039.com/users/4242011-11-12T16:15:29Z2019-04-01T09:51:54Z
<blockquote>
<p>For which values of $n$ does there exist an embedding of a smooth compact manifold $M\hookrightarrow R^n$ into $n$-dimensional Euclidean space such that the normal bundle to $M$ has no nonvanishing section? </p>
<p>If such an embedding does exist, can $M$ be taken to be orientable? What can be the dimension of $M$?</p>
</blockquote>
<p>In other words, I'm asking for submanifolds $M$ of $R^n$ which cannot be "immediately pushed off of themselves" by the flow of a vector field. Conversely, a theorem <a href="http://www.ams.org/journals/tran/1959-093-02/S0002-9947-1959-0119214-4/" rel="noreferrer">of Hirsch</a> states that if a nonvanishing normal field does exist then $M$ immerses in $R^{n-1}$.</p>
<p>If $n=4k$ for some integer $k$, then one can find <i>non-orientable</i> embedded $2k$-submanifolds $M$ in $R^{4k}$ with which have no such sections---the easiest (to me) way that I know of doing this is to ask for $M$ to be Lagrangian with respect to the standard symplectic structure, so that its normal bundle will be isomorphic to its tangent bundle and so one just needs $M$ to have nonzero Euler characteristic (since then the (integral) twisted Euler class in the local coefficient cohomology associated to $w_1$ will be nontrivial), and then examples can be constructed by <a href="http://www.springerlink.com/content/v81010g384q30028/" rel="noreferrer">Lagrangian surgery</a>. On the other hand an orientable submanifold of $R^n$ of dimension $\frac{n}{2}$ will always have a nonvanishing normal vector field, since the only obstruction to constructing such is the Euler class of the normal bundle, which is the restriction of a cohomology class from $R^n$ and so vanishes.</p>
<p>If $\dim M>\frac{n}{2}$ then there are higher-order obstructions to the existence of a nonvanishing section of the normal bundle, which can be nontrivial in the orientable case. There are some examples of this described <a href="http://www.ams.org/journals/proc/1961-012-01/S0002-9939-1961-0124914-0/" rel="noreferrer">by Massey</a>; for instance for $s\geq 1$, $CP^{2^{s}}$ embeds in $R^{4\cdot2^s-1}$, and there is no normal section because the secondary obstruction is nontrivial. I think (though haven't checked carefully) that a product of two of these $CP^{2^s}$ embeddings should also lack a nonvanishing normal vector field, due to product formulas for the obstruction classes. But these examples seem rather special, and I haven't been able to see a way of adapting the ideas underlying them to construct examples in arbitrary ambient dimension.</p>
http://www.4124039.com/q/25966446Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedraDylan Thurstonhttp://www.4124039.com/users/50102017-01-15T16:36:28Z2019-03-31T14:52:07Z
<p>Suppose you have a tetrahedron <span class="math-container">$T$</span> in Euclidean space with edge lengths <span class="math-container">$\ell_{01}$</span>, <span class="math-container">$\ell_{02}$</span>, <span class="math-container">$\ell_{03}$</span>, <span class="math-container">$\ell_{12}$</span>, <span class="math-container">$\ell_{13}$</span>, and <span class="math-container">$\ell_{23}$</span>. Now consider the tetrahedron <span class="math-container">$T'$</span> with edge lengths
<span class="math-container">$$\begin{aligned}
\ell'_{02} &= \ell_{02} &
\ell'_{13} &= \ell_{13}\\
\ell'_{01} &= s-\ell_{01} &
\ell'_{12} &= s-\ell_{12}\\
\ell'_{23} &= s-\ell_{23}&
\ell'_{03} &= s-\ell_{03}
\end{aligned}
$$</span>
where <span class="math-container">$s = (\ell_{01} + \ell_{12} + \ell_{23} + \ell_{03})/2$</span>.
If the edge lengths of <span class="math-container">$T'$</span> are positive and satisfy the triangle inequality, then the volume of <span class="math-container">$T'$</span> equals the volume of <span class="math-container">$T$</span>. In particular, if <span class="math-container">$T$</span> is a flat tetrahedron in <span class="math-container">$\mathbb{R}^2$</span>, then <span class="math-container">$T'$</span> is as well. This is easily verified by plugging the values <span class="math-container">$\ell'_{ij}$</span> above into the Cayley-Menger determinant.</p>
<p>In fact, it's possible to show that the linear symmetries of <span class="math-container">$\mathbb{R}^6$</span> that preserve the Cayley-Menger determinant form the Weyl group <span class="math-container">$D_6$</span>, of order <span class="math-container">$2^5 * 6! = 23040$</span>. This is a factor of <span class="math-container">$15$</span> times larger than the natural geometric symmetries obtained by permuting the vertices of the tetrahedron and negating the coordinates.</p>
<p>The transformations don't always take Euclidean tetrahedra to Euclidean tetrahedra, but they do sometimes. For instance, if you start with an equilateral tetrahedron <span class="math-container">$T$</span> with all side lengths equal to <span class="math-container">$1$</span>, then <span class="math-container">$T'$</span> is also an equilateral tetrahedron. Thus if <span class="math-container">$T$</span> is a generic Euclidean tetrahedron close to equilateral, <span class="math-container">$T'$</span> will also be one, and <span class="math-container">$T$</span> and <span class="math-container">$T'$</span> will not be related by a Euclidean symmetry.</p>
<p>I can't be the first person to observe this. (In fact, I vaguely recall hearing about this in the context of quantum groups and the Jones polynomial.) What's the history? How to best understand these transformations (without expanding out the determinant)? Are <span class="math-container">$T$</span> and <span class="math-container">$T'$</span> scissors congruent? Etc.</p>
http://www.4124039.com/q/3266203Hyperbolic 3 manifold with trivial deformation of flat conformal structuresGorapada Berahttp://www.4124039.com/users/1177232019-03-28T21:25:15Z2019-03-29T18:34:25Z
<p>Does there exist closed hyperbolic three manifold which is locally rigid in the space of its all conformally flat structures? If so could someone provide examples?</p>
http://www.4124039.com/q/3126353Sphere spectrum, Thom spectrum, and Madsen-Tillmann bordism spectrumwonderichhttp://www.4124039.com/users/270042018-10-12T02:58:54Z2019-03-27T16:56:34Z
<p>This is a following up question of <a href="http://www.4124039.com/q/312618/27004">Sphere spectrum, Character dual and Anderson dual</a>.</p>
<p>What are the differences and the significances of the following: </p>
<p>(1). Homotopy classes of maps from a <a href="https://link.springer.com/content/pdf/10.1007/BF02566923.pdf" rel="nofollow noreferrer"><strong>Thom spectrum</strong></a> to a shift of the Anderson dual to the sphere spectrum?</p>
<p>(2). Homotopy classes of maps from a <a href="https://projecteuclid.org/euclid.acta/1485892397" rel="nofollow noreferrer"><strong>Madsen-Tillmann bordism spectrum</strong></a> to a shift of the Anderson dual to the sphere spectrum?</p>
<p>It looks to me that <a href="https://projecteuclid.org/euclid.acta/1485892397" rel="nofollow noreferrer">Madsen-Tillmann bordism spectrum</a> is a close relative of
<a href="https://link.springer.com/content/pdf/10.1007/BF02566923.pdf" rel="nofollow noreferrer">Thom spectrum</a>. So what will be the comparison, differences, similarity between the two Homotopy classes above?</p>
<p>p.s. I suppose my Journal Article links above/below for two spectra are 100% correct. Please correct me if I am imprecise or I miss the Refs.</p>
<ul>
<li><p>R. Thom, Commentarii Mathematici Helvetici 28, 17 (1954).</p></li>
<li><p>S. Galatius, I. Madsen, U. Tillmann, and M. Weiss, Acta
Math. 202, 195 (2009)</p></li>
</ul>
http://www.4124039.com/q/3264934Hilbert space compression of lamplighter over lamplighter groupsARGhttp://www.4124039.com/users/189742019-03-27T15:33:48Z2019-03-27T15:33:48Z
<p><span class="math-container">$C_2 \wr \mathbb{Z}$</span> is the lamplighter group but I'm currently looking at the lamplighter group with this group as a base space.</p>
<p><b>Question:</b> Consider the group <span class="math-container">$C_2 \wr (C_2 \wr \mathbb{Z})$</span>, what is its compression exponent?</p>
<p>Note that there is a Cayley graph of the group <span class="math-container">$F \wr (F \wr \mathbb{Z})$</span> (where <span class="math-container">$F$</span> is some finite group) which embeds isometrically into
(some Cayley graph of) <span class="math-container">$\mathbb{Z} \wr (\mathbb{Z} \wr \mathbb{Z})$</span>. The latter has compression <span class="math-container">$\tfrac{4}{7}$</span> (a result of Naor and Peres). So the former has at least this exponent (and I'm inclined to think it has exactly this exponent).</p>
<p>On the other hand <span class="math-container">$\mathbb{Z} \wr \mathbb{Z}$</span> has compression exponent <span class="math-container">$\tfrac{2}{3}$</span> while <span class="math-container">$F \wr \mathbb{Z}$</span> has compression exponent 1. So there might be a discrepancy.</p>
<p>Note that the results of Naor & Peres would give an upper bound. But I could not find an estimate on the speed (or drift) of the random walk on <span class="math-container">$C_2 \wr (C_2 \wr \mathbb{Z})$</span></p>
<p><b>Sub-Question:</b> Has the random walk on the group <span class="math-container">$C_2 \wr (C_2 \wr \mathbb{Z})$</span> the same speed/drift (up to logarithmic factors) as the random walk on <span class="math-container">$\mathbb{Z} \wr (\mathbb{Z} \wr \mathbb{Z})$</span>? If not, what is its speed/drift?</p>
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