By Selman Akbulut

This ebook offers the topology of tender 4-manifolds in an intuitive self-contained manner, constructed over a few years by means of Professor Akbulut. The textual content is aimed toward graduate scholars and specializes in the educating and studying of the topic, giving an instantaneous method of structures and theorems that are supplemented through workouts to aid the reader paintings during the information no longer coated within the proofs.

The booklet incorporates a hundred color illustrations to illustrate the tips instead of delivering long-winded and in all likelihood uncertain factors. 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 built in previous chapters.

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**Additional info for 4-Manifolds**

**Example text**

Let Yˆ → Y be the maximal abelian covering of Y , corresponding to the kernel of the natural homomorphism π1 (X) → H, where H is the free abelian group which is the quotient of H1 (Y ) by its torsion subgroup. 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 .

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. 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 .

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]). 7 Some classical invariants Let K ⊂ S 3 be a link. Any oriented surface F ⊂ S 3 bounding K is called a Seifert surface of K. g. [Ro], [AMc], [Liv]), L ∶ H1 (F ; Z) × H1 (F ; Z) → Z Then deﬁne an Alexander polynomial by AF (t) = det(V − tV T ).