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 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 staff and protecting areas, homology and cohomology, better homotopy teams, and homotopy thought often. the writer emphasizes the geometric elements of the topic, which is helping scholars achieve instinct. a different characteristic is the inclusion of many non-compulsory issues now not frequently a part of a primary direction 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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To deduce this from van Kampen’s theorem, choose for each edge eα of X − T an open neighborhood Aα of T ∪ eα in X that deformation retracts onto T ∪ eα . The intersection of two or more Aα ’s deformation retracts onto T , hence is contractible. The Aα ’s form a cover of X satisfying the hypotheses of van Kampen’s theorem, and since the intersection of 44 Chapter 1 The Fundamental Group any two of them is simply-connected we obtain an isomorphism π1 (X) ≈ ∗α π1 (Aα ) . Each Aα deformation retracts onto a circle, so π1 (X) is free on five generators, as claimed.

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. Conversely, if there is only one homotopy class of paths connecting a basepoint x0 to itself, then all loops at x0 are homotopic to the constant loop and π1 (X, x0 ) = 0 . 1 29 The Fundamental Group of the Circle Our first real theorem will be the calculation π1 (S 1 ) ≈ Z . Besides its intrinsic interest, this basic result will have several immediate applications of some substance, and it will be the starting point for many more calculations in the next section.

Every continuous map h : D 2 →D 2 has a fixed point, that is, a point x with h(x) = x . 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 .

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