Download Algebraic Topology: A First Course (Graduate Texts in by William Fulton PDF

By William Fulton

This publication introduces the $64000 principles of algebraic topology by means of emphasizing the relation of those rules with different components of arithmetic. instead of identifying one viewpoint of contemporary topology (homotropy thought, axiomatic homology, or differential topology, say) the writer concentrates on concrete difficulties in areas with a couple of dimensions, introducing in basic terms as a lot algebraic equipment as precious for the issues encountered. This makes it attainable to work out a greater variety of significant good points within the topic than is usual in introductory texts; it's also in concord with the ancient improvement of the topic. The publication is aimed toward scholars who don't inevitably intend on focusing on algebraic topology.

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We say that “pullbacks exist” in the category C if a pullback exists for each pair of morphisms in C with the same target. 1 Example (Pullbacks exist in Set) functions. Define Let λi : Ai → B (i = 1, 2) be two A1 ×B A2 = {(a1 , a2 ) | ai ∈ Ai and λ1 (a1 ) = λ2 (a2 )} ⊆ A1 × A2 , called the fibered product of λ1 and λ2 . Put P = A1 ×B A2 and for i = 1, 2 define σi : P → Ai by σi ((a1 , a2 )) = ai . For a = (a1 , a2 ) ∈ P we have λ1 σ1 (a) = λ1 (a1 ) = λ2 (a2 ) = λ2 σ2 (a). Therefore, λ1 σ1 = λ2 σ2 , implying that the pair (P, (σ1 , σ2 )) is an object of the auxiliary category D = Dpb defined as above (with C = Set).

If λ1 , λ2 : a → b are morphisms in C and (q, π) is a coequalizer of λ1 and λ2 , then π is epic. Proof. 4 with arrows reversed. 7 Pullback Let C be a category and let λi : ai → b (i = 1, 2) be two morphisms in C: a2 a1  λ1 λ2 / b. Form an auxiliary category D = Dpb as follows: Take for objects pairs (x, (α1 , α2 )), where x is an object of C and αi : x → ai (i = 1, 2) are morphisms in C such that λ1 α1 = λ2 α2 , that is, such that the following diagram is commutative: α2 / a2 x α1   a1 λ1 λ2 / b; take as morphisms from the object (x, (α1 , α2 )) to the object (y, (β1 , β2 )) all morphisms γ : x → y in C such that βi γ = αi (i = 1, 2), that is, such that 32 the following diagram is commutative: x α2 γ y α1 β2 β1 λ2    a1 / # a2 λ1 / b; and define composition of morphisms to be the composition in C.

For each a ∈ A, τ1 λ1 (a) = λ1 (a) = λ2 (a) = τ2 λ2 (a) (using that λ1 (a) ∼ λ2 (a)), so τ1 λ1 = τ2 λ2 implying that (Q, (τ1 , τ2 )) is an object of the auxiliary category 38 D = Dpo defined above (with C = Set). Let (X, (α1 , α2 )) be an object of D. Define γ0 : B → X by α1 (b), b ∈ B1 , α2 (b), b ∈ B2 γ0 (b) = (well-defined since B1 and B2 are disjoint). If (r1 , r2 ) ∈ R, then r1 = λ1 (a) and r2 = λ2 (a) for some a ∈ A, so γ0 (r1 ) = α1 λ1 (a) = α2 λ2 (a) = γ0 (r2 ). It follows that γ0 is constant on the equivalence classes of B and therefore induces a unique map γ : Q → X such that γπ = γ0 .

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