By Selman Akbulut
This publication provides the topology of tender 4-manifolds in an intuitive self-contained manner, built over a few years by way of Professor Akbulut. The textual content is geared toward graduate scholars and makes a speciality of the instructing and studying of the topic, giving an instantaneous method of buildings and theorems that are supplemented through workouts to aid the reader paintings throughout the information now not lined within the proofs.
The publication features a hundred color illustrations to illustrate the information instead of delivering long-winded and in all probability uncertain reasons. Key effects were chosen that relate to the cloth mentioned and the writer has supplied examples of the way to examine them with the ideas built in prior chapters.
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Extra info for 4-Manifolds
Additivity of the signature) Let M 4 = M1 ⌣Y M2 be a compact oriented smooth manifold consisting of a union of two compact smooth manifolds glued along their common boundary Y 3 ; then σ(M) = σ(M1 ) + σ(M2 ). This is because the classes of H2 (M) come in three types: (1) surfaces contained entirely in M1 or M2 , (2) surfaces intersecting Y transversally along homologically nontrivial loops C ⊂ Y , (3) surfaces contained in Y 3 . The Poincare duals of type (2) classes (the dual surfaces to the loops C in Y ) are type (3) classes.
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 ).
If K ⊂ Y , we will denote the 3-manifold obtained by surgering Y along K with framing r by the notations Y (K, r) or Yr (K) (provided the framing r in Y is well deﬁned). g. 5). 1) and observing the feet of 1- and 2-handles, we can visualize 2 and 3-manifolds by placing ourselves on the boundary of their 0-handles. 2; except in this case we don’t need to specify the framings of the attaching circles of the 2-handles. The 3-manifold handlebody pictures obtained this way are called Heegaard diagrams.