By Allen Hatcher

"In so much significant universities one of many 3 or 4 uncomplicated 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 assurance and a readable exposition, with many examples and routines. The 4 major chapters current the fundamentals: primary team and protecting areas, homology and cohomology, larger homotopy teams, and homotopy conception often. the writer emphasizes the geometric elements of the topic, which is helping scholars achieve instinct. a distinct characteristic is the inclusion of many non-compulsory themes now not often 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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**Example text**

11. Whenever S 2 is expressed as the union of three closed sets A1 , A2 , and A3 , then at least one of these sets must contain a pair of antipodal points {x, −x} . Proof: Let di : S 2 →R measure distance to Ai , that is, di (x) = inf y∈Ai |x − y| . This is a continuous function, so we may apply the Borsuk–Ulam theorem to the map S 2 →R2 , x d1 (x), d2 (x) , obtaining a pair of antipodal points x and −x with d1 (x) = d1 (−x) and d2 (x) = d2 (−x) . If either of these two distances is zero, then x and −x both lie in the same set A1 or A2 since these are closed sets.

The formula Lgg = Lg Lg implies that Lg is invertible with inverse Lg −1 . Therefore the association g Lg de- fines a homomorphism from Gα to the group P (W ) of all permutations of W . More generally, we can define L : W →P (W ) by L(g1 ··· gm ) = Lg1 ··· Lgm for each reduced word g1 ··· gm . This function L is injective since the permutation L(g1 ··· gm ) sends the empty word to g1 ··· gm . The product operation in W corresponds under L to 42 Chapter 1 The Fundamental Group composition in P (W ) , because of the relation Lgg = Lg Lg .

Proof: The idea is to compare paths in S 1 with paths in R via the map p : R→S 1 given by p(s) = (cos 2π s, sin 2π s) . This map can be visualized geometrically by embedding R in R3 as the helix parametrized by s (cos 2π s, sin 2π s, s) , and then p is the restriction to the he- lix of the projection of R3 onto R2 , (x, y, z) (x, y) , as in the figure. Observe that the loop ωn is the composition pωn where ωn : I →R is the path ωn (s) = ns , starting at 0 and ending at n , p winding around the helix |n| times, upward if n > 0 and downward if n < 0 .