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Stably Complex Structures on Self-Intersection Manifolds of Immersions

Longdon, Alexander

[Thesis]. Manchester, UK: The University of Manchester; 2015.

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Abstract

In this thesis we study the problem of determining the possible cobordism types of r-fold self-intersection manifolds associated to self-transverse immersions f: M^{n-k} -> \R^n for certain values of n, k, and r. Namely, we study the double-point self-intersection 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 self-transverse immersions f : M^{n+2} -> \R^{2n+2}, we see that when n is even the double-point self-intersection manifold is a boundary, which is a result originally due to Szucs. In the case of self-transverse immersions f : M^{n+4} -> \R^{2n+4}, we show than when n is even the double-point self-intersection 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 double-point self-intersection manifold of a self-transverse 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: self-transverse 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 double-point self-intersection 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 self-transverse complex immersions f : M^{2k+2} -> \R^{4k+2}, we show that the first normal Chern number of the double-point self-intersection 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 self-transverse complex immersions f : M^{2k+4} -> \R^{4k+4}, we identify a large class of stably complex manifolds that may arise as the double-point self-intersection 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.

Bibliographic metadata

Type of resource:
Content type:
Form of thesis:
Type of submission:
Degree type:
Doctor of Philosophy
Degree programme:
PhD Mathematical Sciences
Publication date:
Location:
Manchester, UK
Total pages:
204
Abstract:
In this thesis we study the problem of determining the possible cobordism types of r-fold self-intersection manifolds associated to self-transverse immersions f: M^{n-k} -> \R^n for certain values of n, k, and r. Namely, we study the double-point self-intersection 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 self-transverse immersions f : M^{n+2} -> \R^{2n+2}, we see that when n is even the double-point self-intersection manifold is a boundary, which is a result originally due to Szucs. In the case of self-transverse immersions f : M^{n+4} -> \R^{2n+4}, we show than when n is even the double-point self-intersection 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 double-point self-intersection manifold of a self-transverse 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: self-transverse 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 double-point self-intersection 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 self-transverse complex immersions f : M^{2k+2} -> \R^{4k+2}, we show that the first normal Chern number of the double-point self-intersection 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 self-transverse complex immersions f : M^{2k+4} -> \R^{4k+4}, we identify a large class of stably complex manifolds that may arise as the double-point self-intersection 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.
Thesis main supervisor(s):
Thesis co-supervisor(s):
Language:
en

Institutional metadata

University researcher(s):

Record metadata

Manchester eScholar ID:
uk-ac-man-scw:247749
Created by:
Longdon, Alexander
Created:
20th January, 2015, 20:24:36
Last modified by:
Longdon, Alexander
Last modified:
16th November, 2017, 14:24:05

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