By Selman Akbulut

This e-book offers the topology of gentle 4-manifolds in an intuitive self-contained method, built over a couple of years by way of Professor Akbulut. The textual content is aimed toward graduate scholars and makes a speciality of the instructing and studying of the topic, giving a right away method of buildings and theorems that are supplemented by way of workouts to assist the reader paintings in the course of the info no longer coated within the proofs.

The e-book includes a hundred color illustrations to illustrate the guidelines instead of offering long-winded and almost certainly uncertain factors. Key effects were chosen that relate to the fabric mentioned and the writer has supplied examples of ways to examine them with the innovations constructed in prior chapters.

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**Extra info for 4-Manifolds**

**Sample text**

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. Clearly we can thicken these handlebody pictures by crossing them with balls to get higher-dimensional handlebodies, as indicated in these ﬁgures. 2 is a Heegaard diagram of S 3 , and a handlebody of S 3 × B 1 .

Let C∗ = {Ci , ∂} be an acyclic complex ∂ ∂ ∂ Cm → Cm−1 → ... → C0 of ﬁnite-dimensional vector spaces (acyclic means H∗ (C∗ ) = 0). 3). Here, ∂ 2 = 0 implies that this deﬁnition of τ (C∗ ) is independent of the choice of ωi′ s. 3) i=0 We extend this deﬁnition to non-acyclic chain complexes C∗ simply by setting τ (C∗ ) = 0. Let Y be a ﬁnite CW complex and C∗ (Y ) = {Ci (Y ), ∂} be its chain complex, clearly i-cells of Y deﬁne volume on each Ci (Y ). g. closed manifolds are never acyclic). To deﬁne torsion nontrivially we twist the coeﬃcients of C(Y ) so that it will have more chance to be acyclic: For example, we take the universal cover Y˜ → Y , which gives C∗ (Y˜ ) a Z[π1 (Y )]-module structure, then we pick a homomorphism to a ﬁeld λ ∶ Z[π1 (Y )] → F and form a chain complex C∗ (Y ; F ) ∶= C∗ (Y˜ ) ⊗Z[π1 (Y )] F over the ﬁeld F ; we then deﬁne τλ (Y ) = τ (C∗ (Y, F )).

4). Σ(2, 3, 5) is called the Poincare homology sphere. 15. 3. 15, show that Σ(2, 3, 7) ≈ ∂E10 . 16, show that ∂E10 ≈ M 4 , where M 4 is a manifold obtained from E8 by attaching a pair of 2-handles, and M has the intersection form of E8 # (S 2 × S 2 ). 4. 17, construct a closed, simply connected, smooth manifold with signature −16 and the second Betti number 22. 18. 5. 18). 6. 20, show that Σ(2, 3, 11) bounds a deﬁnite manifold with intersection form E8 ⊕ (−1), and also bounds a smooth simply connected manifold with signature −16 and the second Betti number 20.