A General Topology Workbook by Iain T. Adamson

By Iain T. Adamson

This paintings goals to provide undemanding topology in an unconventional approach. It supplies a assessment of the elemental definitions including workouts with out recommendations or proofs of the theorems partly 1, after which provides the ideas partly 2, permitting the scholar to check solutions with their very own.

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Floer Field Philosophy 29 2. To each diffeomorphism φ : Σ0 → Σ1 associate the map Lφ : MΣ0 → MΣ1 which maps ρ ∈ MΣ0 to the representation Lφ (ρ) ∈ MΣ1 given by [γ ] → ρ([φ −1 ◦ γ ]) for any circle γ : S 1 → Σ1 . Observe that Lφ ◦ Lψ = Lψ◦φ when φ, ψ are composable. 3. For each attaching circle α ⊂ Σ we use the bijection πα : Σ α → Σ {2 points} and a deformation of any loop γ : S 1 → Σ to avoid the special points to construct Lα := [ρ], [ρ ] ∈ MΣ− × MΣ ρ([α]) = id, ∀γ : ρ ([γ ]) = ρ([πα−1 ◦ γ ]) .

Z˜g+n )] for x˜ i = πα−1 (xi ), z˜i = πβ−1 (zi ) and some y1 ∈ α, y1 ∈ β. Since α, β are disjoint, this implies y1 = z˜i and y1 = x˜ j for some i, j ≥ 2 which we can permute to i = j = 2 to obtain z2 = πβ (y1 ) ∈ α and x2 = πα (y1 ) ∈ β . Permutation also achieves x˜ i = z˜i for i ≥ 3 and hence xi = πα (yi ), zi = πβ (yi ) for some yi ∈ Σ (α ∪ β), which can be rewritten as φ (πβ (xi )) = πα (zi ) by the defining property of φ applied to yi . While transversality cannot be discussed at the level of sets, note that the intermediate points [ y ] resp.

5) such as HFinst ([0, 1] × Σ, LH0 × LH1 ) HFinst (Y ). 1) is independent of the choice of Heegaard splitting, as required in Step 4. 1. 1. The Heegaard splitting Y = H0− ∪Σ H1 is a decomposition of the morphism [Y ] ∈ (∅, ∅) given by [Y ] = [H0− ] ◦ [H1 ]. Mor Borconn 2+1 2. The representation by symplectic data can be viewed as determining parts of a the functor F : Bor conn 2+1 → Symp by associating to the empty set ∅ ∈ ObjBor conn 2+1 trivial symplectic manifold given by a point F (∅) := pt ∈ ObjSymp , to nonempty the given symplectic manifolds F (Σ) := MΣ , and to surfaces Σ ∈ ObjBorconn 2+1 (∅, Σ) the Lagrangian F (Hi ) := LHi ⊂ pt− × MΣ .

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