By Allen Hatcher

"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 appropriate to be used in a direction at the topic or for self-study, that includes large 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, greater homotopy teams, and homotopy thought in most cases. the writer emphasizes the geometric elements of the topic, which is helping scholars achieve instinct. a different function is the inclusion of many non-compulsory themes now not frequently a part of a primary path because of time constraints: Bockstein and move homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James diminished product, the Dold-Thom theorem, and Steenrod squares and powers."

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The van Kampen theorem will imply that this is indeed the case. Similarly, if X is the union of three circles touching at a single point, the van Kampen theorem will imply that π1 (X) is Z ∗ Z ∗ Z , the group consisting of words in powers of three letters a , b , c . The generalization to a union of any number of circles touching at one point will also follow. The group Z ∗ Z is an example of a general construction called the free product of groups. The statement of van Kampen’s theorem will be in terms of free products, so before stating the theorem we will make an algebraic digression to describe the construction of free products in some detail.

Reparametrizing a path preserves its homotopy class since f ϕ f via the homotopy f ϕt where ϕt (s) = (1 − t)ϕ(s) + ts so that ϕ0 = ϕ and ϕ1 (s) = s . Note that (1 − t)ϕ(s) + ts lies between ϕ(s) and s , hence is in I , so the composition f ϕt is defined. If we are given paths f , g, h with f (1) = g(0) and g(1) = h(0) , then both products (f g) h and f (g h) are defined, and f (g h) is a reparametrization of (f g) h by the piecewise linear function ϕ whose graph is shown in the figure at the right.

In general, a space is called simply-connected if it is path-connected and has trivial fundamental group. The following result explains the name. 6. A space X is simply-connected iff there is a unique homotopy class of paths connecting any two points in X . Proof: Path-connectedness is the existence of paths connecting every pair of points, so we need be concerned only with the uniqueness of connecting paths. Suppose π1 (X) = 0 . If f and g are two paths from x0 to x1 , then f f g g g since the loops g g and f g are each homotopic to constant loops, using the assumption π1 (X, x0 ) = 0 in the latter case.