By Selman Akbulut

This booklet provides the topology of gentle 4-manifolds in an intuitive self-contained means, built over a few years by way of Professor Akbulut. The textual content is aimed toward graduate scholars and makes a speciality of the educating and studying of the topic, giving an immediate method of buildings and theorems that are supplemented by way of routines to assist the reader paintings in the course of the information no longer coated within the proofs.

The publication incorporates a hundred color illustrations to illustrate the information instead of delivering long-winded and probably doubtful motives. Key effects were chosen that relate to the fabric mentioned and the writer has supplied examples of ways to examine them with the ideas built in previous chapters.

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42 Chapter 4 Bundles Here we describe handlebodies of 4-manifolds which are bundles. 2). 1: T 3 4-Manifolds. ©Selman Akbulut 2016. Published 2016 by Oxford University Press. 5 we get another handlebody picture of T 4 . For the benefit of the reader we do this conversion gradually. 5. 1. 2 describes T02 × T 2 , and deleting two 3-handles from T02 × T 2 gives just T02 × T02 , where T02 = T 2 − D 2 is the punctured T 2 . 2 Cacime surface Cacime is a particular surface bundle over a surface, which appears naturally in complex surface theory [CCM].

Cs }, and let {γ1 , . . e. γj = ∂Bj , where Bj is the co-core r of the dual 2-handle of Kj j . Then if {f (γ1 ), . . , f (γk )} is a slice link in N, that is, if each f (γj ) = ∂Dj where Dj ⊂ N are properly imbedded disjoint disks. 22). 22 This reduces the extension problem to the problem of extending f ′ to complements M −M ′ → N −N ′ . Notice that M −M ′ = #s (S 1 ×B 3 ) and N −N ′ is a homotopy equivalent to #s (S 1 × B 3 ). Since every self diffeomorphism #s (S 1 × S 2 ) extends to a unique self diffeomorphism of #s (S 1 × B 3 ), the only way f ′ doesn’t extend to a diffeomorphism M − M ′ → N − N ′ is when N − N ′ is an exotic copy of #s (S 1 × B 3 ).

8. Show K is slice (Hint: perform the slice move along the dotted line indicated in the figure). 11 (c)). 9. 26 the loops a and b are isotopic to each other (Hint: slide them over the 2-handle). From this, produce distinct knots Kr and Lr with Kr0 ≈ Ks0 and L4r ≈ L4s for all r ≠ s (Hint: consider repeated ±1 surgeries to a and b). 2. We call a link L = {K, K+ , . . , K+ , K− , . . , K− }, consisting of K and an even number of oppositely oriented parallel copies of K (pushed off by the framing r), an r-shaking of K.

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