Download e-book for kindle: 4-Manifolds by Selman Akbulut

By Selman Akbulut

ISBN-10: 0198784864

ISBN-13: 9780198784869

This ebook offers the topology of tender 4-manifolds in an intuitive self-contained method, built over a couple of years via Professor Akbulut. The textual content is aimed toward graduate scholars and makes a speciality of the instructing and studying of the topic, giving an immediate method of structures and theorems that are supplemented through workouts to assist the reader paintings during the info no longer coated within the proofs.

The booklet encompasses a hundred color illustrations to illustrate the guidelines instead of supplying long-winded and almost certainly doubtful motives. Key effects were chosen that relate to the cloth mentioned and the writer has supplied examples of the way to examine them with the options constructed in past chapters.

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27). 6 gives an example of pair of knots K1 and K2 , such that one is slice the other is not, and K1−1 is homeomorphic to K2−1 , even though they are not diffeomorphic to each other. So not all knots are shake slice. 10. 7), then for any prime p: ∣σp (L)∣ + np (L) ≤ μ(L) Prove that if ∣σp (K)∣ ≥ 2 and p divides r, then K is not r-shake slice (Hint: show that r-shaking doesn’t change the p-signature, whereas it increases the nullity by 2m where m is the number of pairs of oppositely oriented copies to K ([A1]).

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. Hg is the solid handlebody bounding Fg , and f ∶ Fg → Fg is a diffeomorphism (see Chapter 5). 5. Show that any oriented circle bundle M 4 over Y 3 is a union of two copies of S 1 × Hg glued along their boundaries by a diffeomorphism.

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 ).

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4-Manifolds by Selman Akbulut

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