n-categorical description of Chern classes - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:02:19Z http://mathoverflow.net/feeds/question/117378 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117378/n-categorical-description-of-chern-classes n-categorical description of Chern classes Will 2012-12-28T02:10:13Z 2012-12-28T13:51:47Z <p>The Chern classes of a rank $n$ vector bundle on $X$ are obtained from composing the associated classifying map in $[X, BU(n)]$ with the maps $BU(n) \to B^{2i} \mathbb{Z}$ corresponding to the cohomology of $BU(n)$, thus giving elements in the cohomology groups $H^{2i}(X,\mathbb{Z})$.</p> <p>But one could instead write those maps as $BU(n) \to B^{2i-1} U(1)$; and this corresponds to associating a $(2i-1)$-circle bundle to a vector bundle. For $i = 1$, for instance, this corresponds to sending a vector bundle to its top exterior power, the determinant line bundle.</p> <p>What is then the interpretation of the higher Chern classes in this geometric setting? These should associate to the original rank $n$ vector bundle some circle $(2i-1)$-bundles (e.g. principal $B^{2i-2}U(1)$ $(2i-1)$-bundles), for $1 \leqslant i \leqslant n$.<br> This should mirror the algebraic side with symmetric polynomials; using $e_1 = x_1 + \ldots + x_n$ we have the line bundle $L = L_1 \otimes \ldots \otimes L_n$ (corresponding to the Chern roots); for other elementary symmetric polynomials $e_i$ we should be able to build a corresponding $(2i-1)$-line bundle directly realising the $i$-th Chern class. (Perhaps it would be more natural to instead consider Schur classes, and hopefully the relationship of these $(2i-1)$-line bundles with Schur functors is elucidated.)<br> Is there then an obstruction-theoretic picture, when trying to build back up the original vector bundle from these successive $(2i-1)$-line bundles?</p> http://mathoverflow.net/questions/117378/n-categorical-description-of-chern-classes/117407#117407 Answer by Michael Murray for n-categorical description of Chern classes Michael Murray 2012-12-28T13:51:47Z 2012-12-28T13:51:47Z <p>The answer will depend on your realisation of a $k$-circle bundle. In the case of $i=2$ (second Chern class) there are results associating to any principal $G$-bundle a bundle $2$-gerbe. See: </p> <p>Bundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories. Alan L. Carey, Stuart Johnson, Michael K. Murray, Danny Stevenson and Bai-Ling Wang. Communications in Mathematical Physics, 159 (3) (2005), 577-613 math.DG/0410013</p> <p>and the references there in to Danny Stevenson and Stuart Johnson's PhD theses and papers. Of course you have to be happy that a $3$-circle bundle is a $2$-gerbe.</p> <p>More generally you might find something useful in:</p> <p>P. Gajer Geometry of Deligne cohomology, Invent. Math. 127 (1997), 155-207.</p> <p>which gives a realisation of principal $B^k \mathbb{C}^*$ bundles which are another possible way of realising $(k+1)$-circle bundles or at least mathematical objects determined by a characteristic class in degree $H^{k+1}(M, \mathbb{Z})$. There is a nice inductive classifying theory and a simplicial realisation of these spaces.</p>