By Allen Hatcher

"In such a lot significant universities one of many 3 or 4 simple first-year graduate arithmetic classes is algebraic topology. This introductory textual content is appropriate to be used in a direction at the topic or for self-study, that includes wide insurance and a readable exposition, with many examples and routines. The 4 major chapters current the fundamentals: primary crew and masking areas, homology and cohomology, larger homotopy teams, and homotopy conception commonly. the writer emphasizes the geometric features of the topic, which is helping scholars achieve instinct. a distinct characteristic is the inclusion of many not obligatory issues no longer often a part of a primary path 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 lowered product, the Dold-Thom theorem, and Steenrod squares and powers."

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**Sample text**

If H is a normal subgroup, it implies that G is the direct product of H and the kernel of ρ . If H is not normal, then G is what is called in group theory the semi-direct product of H and the kernel of ρ . Recall from Chapter 0 the general definition of a homotopy as a family ϕt : X →Y , t ∈ I , such that the associated map Φ : X × I →Y , Φ(x, t) = ϕt (x) , is continuous. If ϕt takes a subspace A ⊂ X to a subspace B ⊂ Y for all t , then we speak of a homotopy of maps of pairs, ϕt : (X, A)→(Y , B) .

The idea of continuously deforming a path, keeping its endpoints fixed, is made precise by the following definition. A homotopy of paths in X is a family ft : I →X , 0 ≤ t ≤ 1 , such that (1) The endpoints ft (0) = x0 and ft (1) = x1 f0 are independent of t . (2) The associated map F : I × I →X defined by x0 x1 f1 F (s, t) = ft (s) is continuous. When two paths f0 and f1 are connected in this way by a homotopy ft , they are said to be homotopic. 1: f1 . Linear Homotopies. Any two paths f0 and f1 in Rn having the same endpoints x0 and x1 are homotopic via the homotopy ft (s) = (1 − t)f0 (s) + tf1 (s) .

Here we are using the standard notation D n for the closed unit disk in Rn , all vectors x of length |x| ≤ 1 . Thus the boundary of D n is the unit sphere S n−1 . 32 Chapter 1 The Fundamental Group Proof: Suppose on the contrary that h(x) ≠ x for all x ∈ D 2 . Then we can define a map r : D 2 →S 1 by letting r (x) be the point of S 1 where the ray in R2 starting at h(x) and passing h(x) through x leaves D 2 . Continuity of r is clear since small perturbations of x produce small perturbations of h(x) , hence x r(x) also small perturbations of the ray through these two points.