Topology By: Jarek Esswein.

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Presentation transcript:

Topology By: Jarek Esswein

Problem Suppose that M is a 3-manifold without boundary. Show that the cartesian product M x [0,1] is a 4-manifold. Determine the boundary of M x [0,1].

What is a manifold? What is a 4-manifold? Possesses points with neighborhoods (an open space surrounding point) homeomorphic to R^4 Boundaries of 4-manifolds

Cartesian Products Much like Cartesian products in set theory Every element or topological space is crossed with the other Ex. The cartesian product of A circle (denoted S^(1)) with another circle would be a sphere

Computing Boundaries of Cartesian Products Theorem δ(S x T)= δS x T U S x δT δ denotes boundary Much like the product rule of derivatives but with unions

=(empty set) x [0,1] U M^3 x ({0}U{1}) Problem =M^3 U M^3 =(empty set) x [0,1] U M^3 x ({0}U{1}) =δM^3 x [0,1] U M^3 x δ[0,1] δ(M^3 x [0,1])

Conclusion The Boundary of M^3 x [0,1]= M^3 U M^3 Since the boundary of the manifold is a 3-manifold this implies that the surface itself is a 4-manifold