Download e-book for kindle: Algebraic topology by Morgan J.W., Lamberson P.J.

By Morgan J.W., Lamberson P.J.

Show description

Read or Download Algebraic topology PDF

Best topology books

Homology Theory: An Introduction to Algebraic Topology - download pdf or read online

The two decades because the book of this e-book were an period of continuous development and improvement within the box of algebraic topology. New generations of younger mathematicians were informed, and classical difficulties were solved, relatively during the program of geometry and knot concept.

Download PDF by Kevin Lin, Zhenghan Wang, Weiping Zhang: Topology and Physics: Proceedings of the Nankai

This targeted quantity, as a result of a convention on the Chern Institute of arithmetic devoted to the reminiscence of Xiao-Song Lin, offers a large connection among topology and physics as exemplified by way of the connection among low-dimensional topology and quantum box conception. the quantity comprises works on photograph (2+1) - TQFTs and their functions to quantum computing, categorification and Khovanov homology, Gromov - Witten sort invariants, twisted Alexander polynomials, Faddeev knots, generalized Ricci stream, Calabi - Yau difficulties for CR manifolds, Milnor's conjecture on quantity of simplexes, Heegaard genera of 3-manifolds, and the (A,B) - slice challenge.

Extra info for Algebraic topology

Sample text

Compute the induced map L∗ : Hn (Rn , Rn \ {0}) → Hn (Rn , Rn \ {0}). 6. Let f : Rn → Rn be a diffeomorphism with f (0) = 0. Compute the induced map f∗ : Hn (Rn , Rn \ {0}) → Hn (Rn , Rn \ {0}). 7. 11 we saw that for n ≥ 1 Hk (S n ) = Z 0 k = 0, n otherwise Therefore, if f : S n → S n , the induced map f∗ : Hn (S n ) → Hn (S n ) is multiplication by some integer d. We call this integer the degree of the map f . Compute the degree of the identity map on S n and the antipodal map a : S n → S n . 8.

We need to check that this definition is independent of the closed form representatives that we use. So, if we have two other representatives, α + dγ ∈ [α] and β + dµ ∈ [β], we need to show that, [(α + dγ) ∧ (β + dµ)] = [α ∧ β + α ∧ dµ + dγ ∧ β + dγ ∧ dµ] = [α ∧ β] This follows immediately from the following lemma. 1. The wedge product of a closed from with an exact form is exact (and therefore an exact form wedge a closed form is exact). Proof. e. dα = 0, and dµ is an exact form. Then d(−1)|α| (α ∧ µ) = α ∧ dµ.

For any n-chain ζ ∈ Sn (X) there exists a k so that sdk (ζ) ∈ Snsmall (X). 8. The map on homology induced by S∗small (X) ֒→ S∗ (X) is onto. Proof. Given [ζ] ∈ Hn (X), let ζ be a representative cycle for this homology class. Then sdk (ζ) is also a cycle, and sdk (ζ) is homologus to ζ, but if k is sufficently large sdk (ζ) is in Snsmall (X). 9. If ζ is small, then sd(ζ) and H(ζ) are also small. 10. The map on homology induced by S∗small (X) ֒→ S∗ (X) is injective. Proof. e. [a′ ] = 0 ∈ Hn (X). small (X).

Download PDF sample

Algebraic topology by Morgan J.W., Lamberson P.J.

by Robert

Rated 4.54 of 5 – based on 10 votes