Hatcher - Algebraic Topology



Algebraic Topology

The current version of the text is available at http://www.math.cornell.edu/~hatcher/AT/ATpage.html

"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 Notions=

Homotopy and Homotopy Type
"Decompose a thick letter, call it X, into line segments connecting each point on the outer boundary of X to a unique point of the thin subletter X"

It's not at all obvious that this is actually possible in general.

"It is an easy exercise to check that [homotopy equivalence] is an equivalence relation."

Maybe for someone used to doing topological constructions. If we suppose that we have maps

$$X\underset{f'}\stackrel{f}{\rightleftarrows}Y\underset{g'}\stackrel{g}{\rightleftarrows}Z$$

such that $$f' \circ f \simeq 1_X$$ and $$g' \circ g \simeq 1_Y$$ then this means we have homotopies $$\Phi : X \times I \to X$$ and $$\Gamma : Y \times I \to Y$$ such that $$\Phi_0 = 1_X$$, $$\Phi_1 = f' \circ f$$, $$\Gamma_0= 1_Y$$, $$\Gamma_1 = g' \circ g$$ (along with two more homotopies going in the other direction).

Define $$ H : X \times I \to X$$ by

$$H_t := \begin{cases} f' \circ \Gamma_{2t-1} \circ f && \text{ if } t \geq 1/2; \\ \Phi_{2t} && \text{ if } t < 1/2. \end{cases}$$

we can check that the two branches match up at $$t = 1/2$$ so this is continuous, and so it's a homotopy from $$f' \circ g' \circ g \circ f$$ to $$1_X$$. And we can build a homotopy in the opposite direction as well. So $$g \circ f$$ is a homotopy equivalence of X and Z, with homotopy inverse $$f' \circ g'$$.

"One can in fact take Z to be the mapping cylinder $$M_f$$ of any homotopy equivalence $$f: X \to Y$$"

Careful here because we are doing things in a slightly different context than before. In the example where mapping cylinders were introduced, we were taking a map from the boundary  of a "thick" letter to the underlying "thin" letter. This map was not necessarily a homotopy equivalence, and the resulting mapping cylinder $$M_f$$ turned out to be the "thick" letter itself.

In this case we are starting  with a homotopy equivalence $$f: X \to Y$$. So the example of mapping cylinders we saw before will not be an illustration of what we're doing with mapping cylinders here.

Uncategorized links

 * http://math.stackexchange.com/questions/14483/question-about-the-first-proof-in-hatchers-algebraic-topology
 * http://math.stackexchange.com/questions/70251/question-in-hatcher
 * http://math.stackexchange.com/questions/26179/hatchers-terminology

=Chapter 1: The Fundamental Group=


 * http://math.stackexchange.com/questions/26186/proof-of-another-hatcher-exercise-homotopy-equivalence-induces-bijection

=Chapter 2: Homology=

$$\mathbb{Z} \cong {\rm im} \, \varepsilon \cong C_0(X)/\ker \varepsilon \cong C_0(X) / {\rm im} \, \partial_1 = H_0(X)$$
 * Proposition 2.6:


 * 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)


 * Theorem 2.26: We seem to need {x} closed for this sort of argument to hold.
 * http://math.stackexchange.com/questions/21913/question-about-singular-homology-section-in-hatcher

=Chapter 3: Cohomology=

=Chapter 4: Homotopy Theory=

=Appendix=