Almost Complex Manifolds with Prescribed Betti Numbers

Abstract

The original version of Sullivan's rational surgery realization theorem provides necessary and sufficient conditions for a prescribed rational cohomology ring to be realized by a simply-connected smooth closed manifold. We will present a version of the theorem for almost complex manifolds. It has been shown there exist closed smooth manifolds M^n of Betti number b_i=0 except b_0=b_{n/2}=b_n=1 in certain dimensions n>16, which realize the rational cohomology ring Q[x]/^3 beyond the well-known projective planes of dimension 4, 8, 16. By the obstructions from the signature equation and the Riemann-Roch integrality conditions among Chern numbers, one can show that none of these manifolds with sum of Betti number three in dimension n>4 can admit almost complex structure. More generally, any 4k (k>1) dimensional closed almost complex manifold with Betti number b_i = 0 except i=0,n/2,n must have even signature and even Euler characteristic, one can characterize all the realizable rational cohomology rings by a set of congruence relations among the signature and Euler characteristic.

Speaker

I have been teaching at the University of Washington since autumn 2020. Before coming to UW, I was a senior lecturer at Indiana University Bloomington (2014-2020). I have also taught at University of California Irvine (2012-2014). I was an assistant professor at Rose-Hulman Institute of Technology (2009-2012).

My research interests are in geometric topology and rational homotopy theory. I have been studying existence of high dimensional smooth manifolds realizing prescribed rational cohomology rings. By rational surgery theory, the realization is guaranteed by characteristic numbers satisfying the signature equation and Riemann-Roch integrality relations. I did my PhD thesis, “rational homotopy types of smooth manifolds”, under the supervision of James Davis at Indiana University in 2009.

home page：

https://sites.uw.edu/zhixusu/