Knot theory and 3-manifolds.

Contents: definition of knots and links. asphericity of knot ( and irreducible link ) complement using Sphere theorem. boundary incompressiblity of knot complement. characterization of trivial knot in terms of its fundamental group. definition and existance of Seifert surfaces of knots and links. definition of genus of knots. incompressibility of minimal Seifert surface. additivity property of genus on sum of knots. existance of infinitely many knots. construction of infinite cyclic cover using minimal Seifert surface of a knot. definition of Dehn twist and proving that the oriented mapping class group of any closed oriented surface is generated by Dehn twists. definition of Dehn surgery along a knot or link. proving that any closed 3-manifold is obtained by Dehn surgery along a link in the 3-sphere. showing that any two closed 3-manifolds are cobordant. definition of Jones polynomial of links and some properties. and many exercises and problems. (3 lectures ).

Seifert fibered spaces .

Contents: definition and examples of Seifert fibered spaces. finite sheeted coverings of Seifert fibered spaces are Seifert fibered. irreducibility of Seifert fibered spaces with nonempty boundary. presentation of fundamental group of Seifert fibered spaces. Haken 3-manifolds with an infinite cyclic normal subgroup of the fundamental group are either Seifert fibered or union of two twisted I-bundles along their boundary. and many exercises. (1 lecture ).

 Simple homotopy types of 4-manifolds .

Contents: simple homotopy equivalence, Whitehead group, s-cobordism theorem, simple homotopy types up to s-cobordism of 4-manifolds. (1 lecture ).

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