Calculus

Download 4-Manifolds by Selman Akbulut PDF

By Selman Akbulut

ISBN-10: 0198784864

ISBN-13: 9780198784869

This e-book offers the topology of delicate 4-manifolds in an intuitive self-contained approach, constructed over a couple of years via Professor Akbulut. The textual content is geared toward graduate scholars and specializes in the instructing and studying of the topic, giving an immediate method of structures and theorems that are supplemented via workouts to assist the reader paintings during the information now not lined within the proofs.

The ebook includes a hundred color illustrations to illustrate the guidelines instead of supplying long-winded and probably doubtful motives. Key effects were chosen that relate to the fabric mentioned and the writer has supplied examples of the way to examine them with the ideas constructed in previous chapters.

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Example text

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. We call a knot K ⊂ S 3 an r-shake slice if an r-shaking of K bounds a disk with holes in B 4 . The r-shake genus of a knot K is the smallest integer g where some r-shaking of K occurs as a boundary of a genus g surface with holes in B 4 .

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 ).

H acts on Y˜ by deck transformations making C∗ (Y˜ ) a Z[H]-module. Z[H] is an integral domain (because it is UFD ([Tan]) so we can take its field of fractions F = Q[H], which we can consider a Z[H]-module. Hence we can form a chain complex C(Y, F ) ∶= C∗ (Y˜ ) ⊗Z[π1 (Y )] F over the field F . Milnor torsion is defined to be as the torsion of this chain complex ν(Y ) ∈ F /H. 7. If K ⊂ S 3 is a knot, and Y = SK is the 3-manifold obtained by surgering S 3 along K with framing 0, then H = Z and Z[H] = Z[t, t−1 ].

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