By Selman Akbulut

This e-book offers the topology of soft 4-manifolds in an intuitive self-contained means, built over a couple of years through Professor Akbulut. The textual content is aimed toward graduate scholars and specializes in the instructing and studying of the topic, giving an instantaneous method of structures and theorems that are supplemented via routines to aid the reader paintings during the info no longer lined within the proofs.

The booklet encompasses a hundred color illustrations to illustrate the tips instead of supplying long-winded and probably doubtful factors. Key effects were chosen that relate to the cloth mentioned and the writer has supplied examples of ways to examine them with the thoughts constructed in prior chapters.

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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.