Title:

Stably complex structures on selfintersection manifolds of immersions

In this thesis we study the problem of determining the possible cobordism types of rfold selfintersection manifolds associated to selftransverse immersions f: M^{nk} > \R^n for certain values of n, k, and r. Namely, we study the doublepoint selfintersection manifolds of immersions M^{n+2} > \R^{2n+2} and M^{n+4} > \R^{2n+4}, focusing on the case when $n$ is even. In the case of selftransverse immersions f : M^{n+2} > \R^{2n+2}, we see that when n is even the doublepoint selfintersection manifold is a boundary, which is a result originally due to Szucs. In the case of selftransverse immersions f : M^{n+4} > \R^{2n+4}, we show than when n is even the doublepoint selfintersection manifold is either a boundary or cobordant to RP^2 x RP^2, which is a new result. We then show that for even n such that the binary expansion of n+4 contains 5 or more 1s, the doublepoint selfintersection manifold of a selftransverse immersion M^{n+4} > \R^{2n+4} is necessarily a boundary. We also survey the case when n is odd. We also set up and study the complex versions of the above problems: selftransverse immersions f : M^{2k+2} > \R^{4k+2} and f : M^{2k+4} > \R^{4k+4} of stably complex manifolds with a given complex structure on the normal bundle of f$. In these cases, the doublepoint selfintersection manifold L associated to the immersion inherits a stably complex structure, and we attempt to determine which complex cobordism classes of stably complex manifolds may arise in this way. This is all new work. In the case of selftransverse complex immersions f : M^{2k+2} > \R^{4k+2}, we show that the first normal Chern number of the doublepoint selfintersection manifold is a multiple of 2^{\lambda_{k+1}} for some integer \lambda_{k+1}, and provide upper and lower bounds for the value of \lambda_{k+1}. We also determine the exact value of \lambda_{k+1} in certain cases. In the case of selftransverse complex immersions f : M^{2k+4} > \R^{4k+4}, we identify a large class of stably complex manifolds that may arise as the doublepoint selfintersection manifold of such an immersion and also identify a class of manifolds that may not. Additionally, in both cases we identify a necessary (and sometimes sufficient) condition for a stably complex manifold of the appropriate dimension to admit a complex immersion of the appropriate codimension.
