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11. 8. To construct E ′ we simply cross this twisted F2 bundle over the circle F2 ×τ S 1 with S 1 . 9. 4. 14 describes E0′ = E ′ −F2 ×D 2 . 15. 15. 16. 17. 16 by f2−1 ○ f1 . 3 General surface bundles over surfaces Now it is clear how to proceed in drawing a handlebody picture of a general Fg bundle ˜ Fp . 18. By removing Fg × D 2 from each, we write M =⌣∂ Ej , where each Ej is a Fg bundle over T02 , then we perform gluing operations along the boundaries, as described in Chapter 3. 2 p Fg ... e.
1). In the special case of when M has no 3-handles, then clearly the framed link {f (γ1), . . , f (γn )} in ∂B 4 gives its upside down handlebody of M. 4-Manifolds. ©Selman Akbulut 2016. Published 2016 by Oxford University Press. 1 The manifold −M ⌣id M is called the double of M and denoted by D(M). So by the above, D(M) is a handlebody obtained from M by attaching 2-handles along the zero-framed dual circles of the 2-handles of M. This gives D(T 2 × B 2 ) = T 2 × S 2 . The cusp is defined to be the 4-manifold K 0 , where K is the right handed trefoil knot.
Co]). For example L(7, 1) and L(7, 2) are homotopy equivalent manifolds with different torsions. Compact smooth manifolds Y have unique PL-structure, so torsion is a diffeomorphism invariant. There is an other notion of torsion, defined by Milnor, for manifolds with b1 (Y ) > 0, as follows. Let Yˆ → Y be the maximal abelian covering of Y , corresponding to the kernel of the natural homomorphism π1 (X) → H, where H is the free abelian group which is the quotient of H1 (Y ) by its torsion subgroup.