Shadows and branched shadows of 3- and 4-manifolds

Francesco Costantino (Autor)

Buch | Softcover

183 Seiten

2005
Scuola Normale Superiore (Verlag)
978-88-7642-154-9 (ISBN)

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The theory of shadows of 3 and 4-manifolds represents a bridge between combinatorics of polyhedra and low-dimensional topology. On the one hand, it allows a purely combinatorial approach to the study of smooth 4-manifolds and, on the other, it indicates relations between old-standing problems in group theory and recent topological results on 4-dimensional manifolds. The present Ph.D. thesis is devoted to further develop these connections and to find new applications to low-dimensional topology. The results proved, for the most part, seem to strengthen the idea that topology of 3-manifolds can be used as a guide to study the 4-dimensional case and that polyhedra can be used as a “bridge”: in many cases the 4-dimensional results based on shadows restrict through the theory of spines to results about 3-dimensional topology and geometry. On the 3-dimensional side, a new notion of “shadow-complexity” of 3-manifolds is defined. The study of this complexity clarifies how hyperbolic geometry of 3-manifolds is intimately connected with the combinatorial structure of the polyhedra used to describe the manifolds. On the 4-dimensional side, the notion of branched shadow is introduced in order to study, through a purely combinatorial approach, differentiable objects as Spinc and almost complex structures on smooth 4-manifolds. Combinatorial sufficient conditions based on these objects are proved assuring that “refined” structures on 4-manifolds exist such as integrable complex structures and Stein domain structures.

Erscheint lt. Verlag	1.10.2005
Reihe/Serie	Publications of the Scuola Normale Superiore ; 1
Reihe/Serie	Theses (Scuola Normale Superiore)
Zusatzinfo	XX, 183 p.
Verlagsort	Pisa
Sprache	englisch
Maße	150 x 240 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	88-7642-154-8 / 8876421548
ISBN-13	978-88-7642-154-9 / 9788876421549
Zustand	Neuware