Algebraic Topology Allen Hatcher

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Chapter 0

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• Exercise 16 The infinite sphere is contractible

Chapter 2

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• Exercise 1.1 Exhibiting a Mobius strip a a quotient of a two-simplex
• Exercise 1.4 Homology of the triangular parachute
• Exercise 1.6 Another simplicial homology computation
• Exercise 1.7 Exhibiting $S^3$ as a quotient of a tetrahedron
• Exercise 1.8 A simplicial homology computation
• Exercise 1.9 Computing the homology of $\Delta^k$ when all faces of the same dimension are identified
• Exercise 1.12 Chain homotopy of chain maps is an equivalence relation
• Exercise 1.13 Chain homotopic maps remain chain homotopic when considering reduced homology
• Exercise 1.14 Filling in arrows to make a short exact sequence
• Exercise 1.16 The zero’th homology group of $X$ relative to $A$ is trivial precisely when $A$ meets each path component of $X$
• Exercise 1.17 A relative homology calculation for several spaces
• Exercise 1.31 Exhibiting a case of the five-lemma, where the middle isomorphism is non-zero but all other maps are zero.
• Exercise 2.2 Continuous maps $S^{2n} \rightarrow S^{2n}$ have a point which is either fixed, or sent to its antipode
• Exercise 2.3 A non-vanishing vector field on the solid ball has a point where it point radially outward, and another where it points radially inward
• Exercise 2.4 Exhibiting a surjective map $S^n \rightarrow S^n$ of degree zero
• Exercise 2.9 Homology computations for several two-dimensional CW-complxes
• Exercise 2.11 Homology computation for a certain space obtained from a 3-cube under “twisted” face identifications
• Exercise 2.28 Homology computations with Meyer-Vietoris
• Exercise 2.29 Calculating the homology of the interiors of two copies of a 2-surface identified along the 2-surface itself
• Exercise 2.31 Calculating homology of the suspension of a space

Chapter 3

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• Exercise 1.5 Basic proprties of co-chains
• Exercise 1.6 Computing co-homology of basic cell complexes