By Allen Hatcher

ISBN-10: 0521795400

ISBN-13: 9780521795401

"In such a lot significant universities one of many 3 or 4 uncomplicated 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 extensive insurance and a readable exposition, with many examples and workouts. The 4 major chapters current the fundamentals: primary team and overlaying areas, homology and cohomology, larger homotopy teams, and homotopy thought typically. the writer emphasizes the geometric facets of the topic, which is helping scholars achieve instinct. a special function is the inclusion of many not obligatory themes no longer 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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In particular, the commutator aba−1 b−1 is a nontrivial element of this group. The fundamental group of the complement of the two linked circles A and B in the third example is the free abelian group on two generators, represented by the loops a and b linking A and B . As a result of these calculations, we have two ways to tell when a pair of circles A and B is linked. The direct approach is given by the first example, where one circle is regarded as an element of the fundamental group of the complement of the other circle.

Hence (f g) h f (g h) . Restricting atten- tion to loops at the basepoint x0 , this says the product in π1 (X, x0 ) is associative. Given a path f : I →X , let c be the constant path at f (1) , defined by c(s) = f (1) for all s ∈ I . Then f c is a reparametrization of f via the function ϕ whose graph is shown in the first figure at the right, so f c c f f . Similarly, f where c is now the constant path at f (0) , using the reparametrization function in the second figure. Taking f to be a loop, we deduce that the homotopy class of the constant path at x0 is a two-sided identity in π1 (X, x0 ) .

Show this applies in particular when X is open or when X is a union of finitely many closed convex sets. 5. Show that for a space X , the following three conditions are equivalent: (a) Every map S 1 →X is homotopic to a constant map, with image a point. (b) Every map S 1 →X extends to a map D 2 →X . (c) π1 (X, x0 ) = 0 for all x0 ∈ X . Deduce that a space X is simply-connected iff all maps S 1 →X are homotopic. ’] 6. We can regard π1 (X, x0 ) as the set of basepoint-preserving homotopy classes of maps (S 1 , s0 )→(X, x0 ) .

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