User ubunke - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:18:45Z http://mathoverflow.net/feeds/user/7530 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119142/tubular-neighborhoods-of-chains Tubular neighborhoods of chains ubunke 2013-01-17T07:24:40Z 2013-01-28T12:38:53Z <p>A positive answer to the following question would be very helpful in understanding the evaluation of differential cohomology classes on chains.</p> <p>Let $M$ be a smooth manifold and $c$ be a smooth $p$-chain in $M$, i.e. a finite linear combination of smoothly parametrized singular $p$ simplices. Let $|c|\subseteq M$ denote the support of the chain defined as a union of the simplices.</p> <p>Here is the question: Does there exists an open subset $U\subseteq M$ such that $|c|\subseteq U$ and $H^{q}(U,\mathbb{Z})=0$ for all $q\ge p+1$ (real coefficients would be sufficient in the application.</p> http://mathoverflow.net/questions/118548/teaching-stacks-to-differential-geometry-students/118567#118567 Answer by ubunke for Teaching stacks to differential geometry students ubunke 2013-01-10T20:12:51Z 2013-01-10T20:12:51Z <p>I had a good experience with Heinloth's notes. I tried to explain the two-categorical stuff in the example of the stack of principal $G$-bundles. For example, a nice way to understand 2-pull-backs is to calculate $G\cong *\times_{BG}*$ explicitly. And of course, orbifolds and gerbes, e.g. of $Spin^{c}$-reductions of a $Spin^{c}$-principal bundle a provide examples accessible to differential geometers.</p> http://mathoverflow.net/questions/117986/a-lost-lemma-about-periodicity-in-a-grid-of-long-exact-sequences/117995#117995 Answer by ubunke for A lost lemma about periodicity in a grid of long exact sequences? ubunke 2013-01-03T21:21:36Z 2013-01-03T21:21:36Z <p>One application of your lemma is in differential cohomology. See e.g. Ex. 3.25 in <a href="http://arxiv.org/pdf/1208.3961" rel="nofollow">arxiv</a>. </p> <p>I would be very interested in a generalization of this lemma to triangulated categories. So replace your grid of exact sequences by a grid of triangles in a triangulated category. Instead of cohomology you consider the group Hom(T,...) for a fixed object T. Then you get similar long exact sequence and can state an analogous lemma. Is there a proof in this generality?</p> http://mathoverflow.net/questions/83349/a-p-form-taking-discrete-values-on-p-chains-must-be-0/85036#85036 Answer by ubunke for A p-form taking discrete values on p-chains must be 0. ubunke 2012-01-06T08:10:28Z 2012-01-06T08:10:28Z <p>You can determine the value of the $p$-form at a point as the limit if integrals over very small $p$-simplices and rescaling. If the integral takes values in a discrete subgroup of $\mathbb{R}$, then you get zero.</p> http://mathoverflow.net/questions/82376/twisted-de-rham-cohomology-and-eilenberg-mac-lane-spaces/82435#82435 Answer by ubunke for Twisted de-Rham cohomology and Eilenberg-Mac Lane spaces ubunke 2011-12-02T07:39:44Z 2011-12-10T10:17:52Z <p>You should turn your d+H into a flat superconnection of degree one. Here is the example which works for K-theory. You consider the $\mathbb{Z}$-graded complex $\Omega(M)[b,b^{-1}]$ with $b$ of degree $-2$ and define the superconnection by $d+bH$ for the closed $3$-form $H$. There is an equivalence between the $\infty$-categories of such superconnections and bundles of chain complexes (representations of the singular complex of your underlying manifold), see <a href="http://arxiv.org/abs/0908.2843" rel="nofollow">Block-Smith</a>. If you then apply the Eilenberg-MacLane equivalence between the categories of chain complexes and $H\mathbb{Z}$-modules, then you get a bundle of $H\mathbb{Z}$-modules. This (or the bundle of its $\infty$-loop spaces) is what you are lokking for. Actually, all these steps are equivalences so that you can go backwards.</p> http://mathoverflow.net/questions/82505/elliptic-genus-for-manifolds-with-boundary/82600#82600 Answer by ubunke for Elliptic genus for manifolds with boundary ubunke 2011-12-04T07:07:35Z 2011-12-04T07:07:35Z <p>This, or a similar variant has been studied in <a href="http://arxiv.org/abs/0912.4875" rel="nofollow">Secondary Invariants for String Bordism and tmf</a> and <a href="http://arxiv.org/abs/0808.0257" rel="nofollow">The f-invariant and index theory</a>. The deviation from modularity caused by the boundary gives the interesting invariant of the boundary.</p> http://mathoverflow.net/questions/119142/tubular-neighborhoods-of-chains Comment by ubunke ubunke 2013-01-29T06:34:06Z 2013-01-29T06:34:06Z You are right. Fact 2.1 of Simons-Sullivan is exactly what I want. Thanks for the hint. http://mathoverflow.net/questions/119142/tubular-neighborhoods-of-chains Comment by ubunke ubunke 2013-01-18T22:17:22Z 2013-01-18T22:17:22Z I need the property for the given chain. http://mathoverflow.net/questions/119142/tubular-neighborhoods-of-chains/119286#119286 Comment by ubunke ubunke 2013-01-18T22:14:59Z 2013-01-18T22:14:59Z I tried similar ideas. Of course I could approximate my $c$ by a very regular chain or current $c^{\prime}$ which then has a neighbourhood as required. But this might be very small and may not contain the original $c$. http://mathoverflow.net/questions/119142/tubular-neighborhoods-of-chains/119264#119264 Comment by ubunke ubunke 2013-01-18T22:10:56Z 2013-01-18T22:10:56Z Matthias, $M$ in your example is not a manifold. There is no chart at zero. http://mathoverflow.net/questions/119142/tubular-neighborhoods-of-chains Comment by ubunke ubunke 2013-01-17T09:50:11Z 2013-01-17T09:50:11Z This is exactly what I have in mind. http://mathoverflow.net/questions/118754/known-size-invariant-for-riemannian-manifolds/118760#118760 Comment by ubunke ubunke 2013-01-13T13:14:40Z 2013-01-13T13:14:40Z Yes, but in the higher dimensional case I suspect that you can make the derivative large at every point without varying the function to much. http://mathoverflow.net/questions/118754/known-size-invariant-for-riemannian-manifolds/118760#118760 Comment by ubunke ubunke 2013-01-13T07:42:25Z 2013-01-13T07:42:25Z Dear Cadoi, have you checked, that your invariant is non-zero?