By Allen Hatcher

ISBN-10: 0521795400

ISBN-13: 9780521795401

"In so much significant universities one of many 3 or 4 simple first-year graduate arithmetic classes is algebraic topology. This introductory textual content is acceptable to be used in a direction at the topic or for self-study, that includes vast insurance and a readable exposition, with many examples and routines. The 4 major chapters current the fundamentals: basic staff and overlaying areas, homology and cohomology, better homotopy teams, and homotopy idea in general. the writer emphasizes the geometric facets of the topic, which is helping scholars achieve instinct. a distinct characteristic is the inclusion of many not obligatory themes now not frequently a part of a primary direction as a result of time constraints: Bockstein and move homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James decreased product, the Dold-Thom theorem, and Steenrod squares and powers."

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When n > 2 we cannot distinguish R2 − {0} from Chapter 1 36 The Fundamental Group Rn − {f (0)} by the number of path-components, but by the preceding calculation of π1 (Rn − {x}) we can distinguish them by their fundamental groups. The more general statement that Rm is not homeomorphic to Rn if m ≠ n can be proved in the same way using either the higher homotopy groups or homology groups. 19 using homology. Induced homomorphisms allow relations between spaces to be transformed into relations between their fundamental groups.

Interestingly, this loop can be knotted, as the figure shows for the case p = 3 , q = 2 . 24. More generally, the n dimensional torus, which is the product of n circles, has fundamental group isomorphic to the product of n copies of Z . This follows by induction on n . Induced Homomorphisms Suppose ϕ : X →Y is a map taking the basepoint x0 ∈ X to the basepoint y0 ∈ Y . For brevity we write ϕ : (X, x0 )→(Y , y0 ) in this situation. Then ϕ induces a homo- morphism ϕ∗ : π1 (X, x0 )→π1 (Y , y0 ) , defined by composing loops f : I →X based at x0 with ϕ , that is, ϕ∗ [f ] = [ϕf ] .

Two basic properties of induced homomorphisms are: (ϕψ)∗ = ϕ∗ ψ∗ for a composition (X, x0 ) → (Y , y0 ) → (Z, z0 ) . ψ ϕ 11∗ = 11 , which is a concise way of saying that the identity map 11 : X →X induces the identity map 11 : π1 (X, x0 )→π1 (X, x0 ) . The first of these follows from the fact that composition of maps is associative, so (ϕψ)f = ϕ(ψf ) , and the second is obvious. These two properties of induced homomorphisms are what makes the fundamental group a functor. 3. If ϕ is a homeomorphism with inverse ψ then ϕ∗ is an isomorphism with inverse ψ∗ since ϕ∗ ψ∗ = (ϕψ)∗ = 11∗ = 11 and similarly ψ∗ ϕ∗ = 11 .

### Algebraic Topology by Allen Hatcher

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