Algebraic Topology

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"This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in spirit, and stays well within the confines of pure algebraic topology. In a sense, the book could have been written thirty or forty years ago since virtually everything in it is at least that old. However, the passage of the intervening years has helped clarify what are the most important results and techniques. For example, CW complexes have proved over time to be the most natural class of spaces for algebraic topology, so they are emphasized here much more than in the books of an earlier generation. This emphasis also illustrates the book's general slant towards geometric, rather than algebraic, aspects of the subject. The geometry of algebraic topology is so pretty, it would seem a pity to slight it and to miss all the intuition it provides."

Chapter 0: Some Underlying Geometric NotionsEdit

Chapter 1: The Fundamental GroupEdit

Chapter 2: HomologyEdit

  • Proposition 2.6:
    $ \mathbb{Z} \cong {\rm im} \, \varepsilon \cong C_0(X)/\ker \varepsilon \cong C_0(X) / {\rm im} \, \partial_1 = H_0(X) $
  • Proposition 2.10: The terms with i = j in the two sums cancel: i.e. the i = j term (top) cancels with the i = j = k + 1 term (below)

Chapter 3: CohomologyEdit

Chapter 4: Homotopy TheoryEdit