By Allen Hatcher

"In so much 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 path at the topic or for self-study, that includes vast assurance and a readable exposition, with many examples and workouts. The 4 major chapters current the fundamentals: primary crew and protecting areas, homology and cohomology, greater homotopy teams, and homotopy concept normally. the writer emphasizes the geometric points of the topic, which is helping scholars achieve instinct. a distinct function is the inclusion of many not obligatory issues now not often 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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Then (c) gives a unique lift F : I × I →R . The restrictions F |{0}× I and F |{1}× I are paths lifting constant paths, hence they must also be constant by the uniqueness part of (a). So ft (s) = F (s, t) is a homotopy of paths, and ft lifts ft since p F = F . We shall prove (c) using just one special property of the projection p : R→S 1 , namely: There is an open cover {Uα } of S 1 such that for each α , p −1 (Uα ) can be (∗) decomposed as a disjoint union of open sets each of which is mapped homeomorphically onto Uα by p .

Construct an explicit homotopy demonstrating this. 11. If X0 is the path-component of a space X containing the basepoint x0 , show that X induces an isomorphism π1 (X0 , x0 )→π1 (X, x0 ) . the inclusion X0 12. Show that every homomorphism π1 (S 1 )→π1 (S 1 ) can be realized as the induced homomorphism ϕ∗ of a map ϕ : S 1 →S 1 . 13. Given a space X and a path-connected subspace A containing the basepoint x0 , show that the map π1 (A, x0 )→π1 (X, x0 ) induced by the inclusion A X is surjective iff every path in X with endpoints in A is homotopic to a path in A .

It remains to prove (a) and (b). Both statements can be deduced from a more general assertion: (c) Given a map F : Y × I →S 1 and a map F : Y × {0}→R lifting F |Y × {0} , then there is a unique map F : Y × I →R lifting F and restricting to the given F on Y × {0} . Statement (a) is the special case that Y is a point, and (b) is obtained by applying (c) with Y = I in the following way. The homotopy ft in (b) gives a map F : I × I →S 1 by setting F (s, t) = ft (s) as usual. A unique lift F : I × {0}→R is obtained by an application of (a).