By Allen Hatcher

ISBN-10: 0521795400

ISBN-13: 9780521795401

"In so much significant universities one of many 3 or 4 easy 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 vast assurance and a readable exposition, with many examples and routines. The 4 major chapters current the fundamentals: basic staff and masking areas, homology and cohomology, greater homotopy teams, and homotopy thought in general. the writer emphasizes the geometric points of the topic, which is helping scholars achieve instinct. a distinct characteristic is the inclusion of many non-compulsory issues 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 lowered product, the Dold-Thom theorem, and Steenrod squares and powers."

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**Example text**

The seven heavily shaded edges form a maximal tree T ⊂ X , a contractible subgraph containing all the vertices of X . We claim that π1 (X) is the free product of five copies of Z , one for each edge not in T . 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 ) .

If X is the union of path-connected open sets Aα each containing the basepoint x0 ∈ X and if each intersection Aα ∩ Aβ is path-connected, then the homomorphism Φ : ∗α π1 (Aα )→π1 (X) is surjective. If in addition each intersection Aα ∩Aβ ∩Aγ is path-connected, then the kernel of Φ is the normal subgroup N generated by all elements of the form iαβ (ω)iβα (ω)−1 , and so Φ induces an isomorphism π1 (X) ≈ ∗α π1 (Aα )/N . 21: Wedge Sums. In Chapter 0 we defined the wedge sum α Xα of a collection of spaces Xα with basepoints xα ∈ Xα to be the quotient space of the disjoint union α Xα in which all the basepoints xα are identified to a single point.

The case n = 1 is easily disposed of since R2 − {0} is path-connected but the homeomorphic space Rn − {f (0)} is not path-connected when n = 1 . When n > 2 we cannot distinguish R2 − {0} from Chapter 1 36 The Fundamental Group Rn − {f (0)} by the number of path-components, but by the preceding calculation of π1 (Rn − {x}) we can distinguish them by their fundamental groups. The more general statement that Rm is not homeomorphic to Rn if m ≠ n can be proved in the same way using either the higher homotopy groups or homology groups.