Vector bundles over Davis-Januszkiewicz spaces with prescribed characteristic classes (Q2884401)

Given a finite simplicial complex \(K\), a space \(DJ(K)\) whose integral cohomology is isomorphic to the Stanley-Reisner algebra of \(K\) is called a Davis-Januszkiewicz space. One way to construct such a space is as the Borel construction of an associated moment-angle complex, as shown by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417--451 (1991; Zbl 0733.52006)], see Theorem 4.8 therein. In the same paper, Davis and Januszkiewicz studied a particular complex vector bundle \(\lambda\) of rank \(m\) over \(DJ(K)\), where \(m\) is the number of vertices of \(K\), and showed that its Chern classes are given by the elementary symmetric polynomials in the generators of the Stanley-Reisner algebra.NEWLINENEWLINEIn the paper at hand, this vector bundle is studied in greater detail. The author shows that if \(K\) is an \((n-1)\)-dimensional finite simplicial complex, then one can split off a trivial vector bundle of rank \(m-n\) from \(\lambda\). It is shown that the remaining bundle \(\xi\) of rank \(n\) is determined by its Chern classes up to isomorphism. A similar statement holds for the realification \(\xi_{\mathbb R}\) and its Pontrjagin and Euler classes. These results are motivated by Davis' and Januszkiewicz' comparison of \(\lambda\) with the bundle given by the Borel construction of the tangent bundle of the moment-angle complex.NEWLINENEWLINETwo applications are given: firstly, isomorphism classes of complex structures on real vector bundles with the same Pontrjagin classes as those of \(\xi_{\mathbb R}\) are counted. Secondly, the existence of an \(r\)-coloring of \(K\) is seen to be equivalent to a splitting property of \(\xi\) (or \(\xi_{\mathbb R}\)).