Download PDF by András I. Stipsicz, Robert E. Gompf: 4-Manifolds and Kirby Calculus (Graduate Studies in

By András I. Stipsicz, Robert E. Gompf

ISBN-10: 0821809946

ISBN-13: 9780821809945

The earlier 20 years have introduced explosive progress in 4-manifold thought. Many books are at present showing that process the subject from viewpoints similar to gauge conception or algebraic geometry. This quantity, in spite of the fact that, deals an exposition from a topological standpoint. It bridges the space to different disciplines and provides classical yet vital topological recommendations that experience now not formerly seemed within the literature. half I of the textual content offers the fundamentals of the speculation on the second-year graduate point and gives an summary of present study. half II is dedicated to an exposition of Kirby calculus, or handlebody idea on 4-manifolds. it really is either effortless and finished. half III bargains intensive a vast diversity of themes from present 4-manifold study. issues contain branched coverings and the geography of complicated surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. purposes are featured, and there are over three hundred illustrations and diverse routines with strategies within the e-book.

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Extra info for 4-Manifolds and Kirby Calculus (Graduate Studies in Mathematics, Volume 20)

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3. We show that Hn (X r ) ∼ = 0 for n > r. Consider the long exact sequence o relative homology → Hn+1 (X i , X i−1 ) → Hn (X i−1 ) → Hn (X i ) → Hn (X i , X i−1 ) → . . 1. In this way, we get a chain of isomorphisms Hn (X r ) ∼ = Hn (X 0 ) . = Hn (X r−1 ) ∼ = ... 4. 1. Let X be a CW complex. 1 Cn (X) := Hn (X n , X n−1 ) ∼ = σ an n-cell ˜ n (Sn ) ∼ H = Z σ an n-cell is a free abelian group. For n < 0, we let Cn (X) be trivial. 2. If X has only finitely many n-cells, then the abelian group Cn (X) is finitely generated.

The boundary is ∂Bc = ∂c + ∂ψn−1 ∂c . Thus Bc is a relative cycle as well. j m be 3. Consider a singular n-chain α = m j=1 λj αj ∈ Sn (X) on X and let Lj for 1 −1 n the Lebesgue numbers for the coverings {αj (Ui ), i ∈ I} of the simplex Δ . Choose k, such that k n n+1 L1 , . . , Lm . Then B k α1 up to B k αm are all chains in SnU (X). Therefore m B k (α) = j=1 λj B k (αj ) =: α ∈ SnU (X). From part 2, we know that B k α is a cycle as well that is homologous to α. 13. For any open covering U, the injective chain map S∗U (X) → S∗ (X) induces an isomorphism in homology, HnU (X) ∼ = Hn (X).

Proof. We prove the claim by induction. μ0 was the difference class [+1] − [−1], and f (0) ([+1] − [−1]) = [−1] − [+1] = −μ0 . We defined μn in such a way that Dμn = μn−1 . Therefore, as D is natural, Hn (f (n) )μn = Hn (f (n) )D−1 μn−1 = D−1 Hn−1 (f (n−1) )μn−1 = D−1 (−μn−1 ) = −μn . 6. The antipodal map A: has degree (−1)n+1 . S n → Sn x → −x 43 Proof. (n) Let fi : Sn → Sn be the map (x0 , . . , xn ) → (x0 , . . , xi−1 , −xi , xi+1 , . . , xn ). 5, one shows that the degree of fi is −1. As A = fn ◦ .

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4-Manifolds and Kirby Calculus (Graduate Studies in Mathematics, Volume 20) by András I. Stipsicz, Robert E. Gompf


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