By Allen Hatcher

ISBN-10: 0521795400

ISBN-13: 9780521795401

"In so much significant universities one of many 3 or 4 simple first-year graduate arithmetic classes is algebraic topology. This introductory textual content is appropriate to be used in a path at the topic or for self-study, that includes wide assurance and a readable exposition, with many examples and workouts. The 4 major chapters current the fundamentals: basic workforce and overlaying areas, homology and cohomology, larger homotopy teams, and homotopy idea mostly. the writer emphasizes the geometric facets of the topic, which is helping scholars achieve instinct. a distinct characteristic is the inclusion of many not obligatory themes no longer 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 lowered product, the Dold-Thom theorem, and Steenrod squares and powers."

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This means that h represents q times a generator of π1 (S 1 ) . Since q is odd, we conclude that h is not nullhomotopic. But h was the composition gη : I →S 2 →S 1 , and η is obviously nullhomotopic in S 2 , so gη is nullhomotopic in S 1 by composing a nullhomotopy of η with g . Thus we have arrived at a contradiction. 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| .

Replacing p by pt in the formula for fr above and letting t go from 1 to 0 , we obtain a homotopy from the loop fr to the loop ωn (s) = e2π ins . 7, ωn represents n times a generator of the infinite cyclic group π1 (S 1 ) . Since we have shown that [ωn ] = [fr ] = 0 , we conclude that n = 0 . Thus the only polynomials without roots in C are constants. Our next application is the Brouwer fixed point theorem in dimension 2 . 9. Every continuous map h : D 2 →D 2 has a fixed point, that is, a point x with h(x) = x .

We write the third and fourth loops as the nega- B a −a b −b tives of the first two since they can be deformed to the first two, but with the opposite orientations, and as we saw in the preceding example, the sum of two oppositely oriented loops is deformable to a trivial loop, not linked with A B a −a b −b anything. We would like to view the expression a + b − a − b as lying in a nonabelian group, so that it is not automatically zero. Changing to the more usual multiplicative notation for nonabelian groups, it would be written aba−1 b−1 , the commutator of a and b .

### Algebraic Topology by Allen Hatcher

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