hatcher 代数拓扑答案版本四（来自中科大）.pdfVIP

Solutions to Algebraic Topology by Allen Hatcher Contents 1 Chapter 0 1 2 Section 1.1 2 3 Section 1.2 3 4 Section 1.3 7 5 Section 1.A 14 6 Section 2.1 16 7 Section 2.2 20 i 1 Chapter 0 Skipped for triviality: 1–3, 9–12, 14–15, 17, 19–22, 24–29. 4. 1 is homotopy inverse for inclusion : . ft 5. Suppose t : is deformation retraction. idX x0 . For each neighborhood , there exists (0 1) s.t. ( ) for all [ 1]. Let = ( ). 0 t 0 t0 = −1 is homotopy from inclusion : to constant map . t t+(1−t)t0 t0 0 6. (a) First deformation retracts to the bottom line [0 1] 0, then deformation retracts to a point. doesn’t deformation retract to any other point because of Exercise 0.5. (b)(c) deformation retracts in the weak sense to the middle zigzag, so it’s a homotopy equivalence. The middle zigzag is homeomorphic t