# Three-Dimensional Topology

## Abstract

By a *closed* manifold we shall mean a compact manifold without boundary. We will use the notation / for the closed interval [0,1]. A compact surface E is called *pants* (or a *pair of pants)* if it is homeomorphic to the sphere with three holes. Suppose that 5 is a closed oriented surface of genus *g.* Remove from *S* interiors of disjoint closed disks *D*_{1}, …, *D*_{ n } We say that *g* is the genus of the resulting surface. The *solid torus* is the product *D*^{2} × S^{1}, where *D*^{2} is the closed 2-disk. The *handlebody* of genus *g* is the 3-manifold that is obtained by attaching *g* solid tori along disjoint boundary disks to the 3-ball. Each handlebody is bounded by the surface of genus *g*. Recall that every closed 3-manifold has zero Euler characteristic; thus for every compact 3-manifold M we have χ(*M*) = x(\(\partial\)*M*)/2. In particular, the pro-jective plane cannot be the boundary of a compact 3-manifold. A 3-manifold is called *aspherical* if it is connected and has contractible universal cover. Suppose that *M* is a compact aspherical 3-manifold; then either \(\partial\)*M* is a single sphere and *M* is contractible, or each boundary component of *M* has nonpositive Euler characteristic (for otherwise the universal cover *M* of *M* would have at least two boundary spheres which implies that \(H_2 (\tilde M) \ne 0\)). Thus, if *M* is compact and aspherical and χ(*M*) = 0, then either *M* is contractible or each component of \(\partial\)M has zero Euler characteristic, i.e., is the torus or the Klein bottle.

## Keywords

Fundamental Group Boundary Component Klein Bottle Solid Torus Incompressible Surface## Preview

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