A planar portrait of a manifold is the pair of the image and the critical
values of the manifold through a stable map into the plane. It can be considerd
a geometric representation of the manifold drawn in the plane. The cusped fan
is its basic local configuration. In this article, we focus on the fibreing
structure over the cusped fan, and give its characterisation. As application,
planar portraits of the real, complex, and quaternion projective plane, regular
toric surfaces, and some sphere bundles over spheres etc. are constructed.
Conversely, the source manifolds of certain planar portraits are detected.Comment: Title, author, and abstract added to the bod

Kobayashi, Mahito

English

arXiv

EPrint Series of Department of Mathematics, Hokkaido University

On the cusped fan in a planar portrait of a manifold

As a way to draw a picture of a manifold in the plane, we consider to take the image of the manifold through a stable map. We call the image, paired with the critical values, the planar portrait of a manifold. The cusped fan is its basic local con guration. In this article, we focus on the  breing structure over the cusped fan, and give its characterisation. As application, the source manifolds of certain planar portraits are characterised, and stable maps of closed manifolds such as the projective plane kP 2 (k = R;C;H), regular complex toric surfaces, and some sphere bundles over spheres etc. are constructed. As a biproduct, we obtain an in nite to one correspondence of projections up to right-left equivalence, of a  xed manifold to a planar portrait. Further applications on characterising manifolds by planar portraits are left to our forthcomming papers

On the cusped fan in a planar portrait of a manifold

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EPrint Series of Department of Mathematics, Hokkaido University

Hokkaido University Collection of Scholarly and Academic Papers