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 D1, …, D n We say that g is the genus of the resulting surface. The solid torus is the product D2 × S1, where D2 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.
KeywordsFundamental Group Boundary Component Klein Bottle Solid Torus Incompressible Surface
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