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By Allan J. Sieradski

This article is an advent to topology and homotopy. subject matters are built-in right into a coherent complete and constructed slowly so scholars are not crushed. the 1st 1/2 the textual content treats the topology of whole metric areas, together with their hyperspaces of sequentially compact subspaces. the second one 1/2 the textual content develops the homotopy type. there are many examples and over 900 routines, representing quite a lot of hassle. This ebook might be of curiosity to undergraduates and researchers in arithmetic.

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Example text

6, more vigilance such as in [26, 27, 53] is required. 1 In order to construct a 2 + 1 connected Floer field theory Bor conn 2+1 → τ Cat, it suffices to construct a functor F : Bor conn 2+1 → Symp that preserves adjunctions. 2 by the following constructions. 24 K. Wehrheim 1. 4). 2. To a diffeomorphism φ : Σ0 → Σ1 associate a symplectomorphism Lφ : MΣ0 → MΣ1 such that Lφ ◦ Lψ = Lφ◦ψ (as maps) when φ, ψ are composable. (Σ, Σ ) between connected 3. 1 associate a Lagrangian submanifold Lα ⊂ MΣ− × MΣ (that is compact and τ -monotone).

L(Ψ |Σn−1 )−1 ◦ LY(n−1)n ◦ LΨ |∂ + Y , L(Ψ |∂ + Y )−1 ◦ L(ι+Y )−1 = Lι−Y , LY01 , LY12 , . . , LY(n−1)n , L(ι+Y )−1 = F ([Y ]). This finishes the proof that the unique extension F is a well-defined functor. 6 below to induce a functor Bor conn d+1 → Cat, as claimed. Here, the existence of the Yoneda functor follows from the fact that Sympτ extends to a 2-category. A formal notion of d + 1 Floer field theory should also include a notion of duality. However, the abstract categorical notion of duality requires a monoidal structure— roughly speaking, an associative multiplication of objects that extends to a bifunctor.

1) 6. For attaching circles α, β ⊂ Σ with transverse intersection in a single point, the composition Yα− ∪Σ Yβ Zφ is diffeomorphic with fixed boundary to the cylindrical cobordism of a diffeomorphism φ : Σα → Σβ determined by φ ◦ πα = πβ on Σ (α ∪ β) and φ(πα (β)) = πβ (α). Ensure that this is reflected by an embedded geometric composition LαT ◦ Lβ = gr(Lφ ). While step 1 fixes the functor F on all objects, steps 2 and 3 fix explicit Lagrangians F ([Y ]) = L Y only for simple morphisms Y as LZφ = Lφ for cylindrical cobordisms, LYα = Lα for 2-handle attachments, and LYα− = LαT for their adjoint 1-handle attachments.

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