By J. P. May

ISBN-10: 0226511820

ISBN-13: 9780226511825

ISBN-10: 0226511839

ISBN-13: 9780226511832

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**Extra info for A Concise Course in Algebraic Topology**

**Example text**

The weak Hausdorff property admits a similar characterization. Lemma. If X is a k-space, then X is weak Hausdorff if and only if ∆X is closed in X × X. 2. The category of compactly generated spaces One major source of point-set level pathology can be passage to quotient spaces. Use of compactly generated topologies alleviates this. Proposition. If X is compactly generated and π : X −→ Y is a quotient map, then Y is compactly generated if and only if (π × π)−1 (∆Y ) is closed in X × X. The interpretation is that a quotient space of a compactly generated space by a “closed equivalence relation” is compactly generated.

Then π ◦ ν = id and id ≃ ν ◦ π since we can define a deformation h : N f × I −→ N f of N f onto ν(X) by setting h(x, χ)(t) = (x, χt ), where χt (s) = χ((1 − t)s). We check directly that ρ : N f −→ Y satisfies the CHP. Consider a test diagram A i0 x A×I g ˜ h x x h G Nf xY ρ G Y. 4. A CRITERION FOR A MAP TO BE A FIBRATION 51 ˜ that makes the We are given g and h such that h ◦ i0 = ρ ◦ g and must construct h diagram commute. We write g(a) = (g1 (a), g2 (a)) and set ˜ t) = (g1 (a), j(a, t)), h(a, where j(a, t)(s) = g2 (a)(s + st) if 0 ≤ s ≤ 1/(1 + t) h(a, s + ts − 1) if 1/(1 + t) ≤ s ≤ 1.

If G has k generators, then χ(B) = 1 − k. If [G : H] = n, then Fb has cardinality n and χ(E) = nχ(B). Therefore 1 − χ(E) = 1 − n + nk. We can extend the idea to realize any group as the fundamental group of some connected space. Theorem. For any group G, there is a connected space X such that π1 (X) is isomorphic to G. 38 GRAPHS Proof. We may write G = F/N for some free group F and normal subgroup N . As above, we may realize the inclusion of N in F by passage to fundamental groups from a cover p : E −→ B.

### A Concise Course in Algebraic Topology by J. P. May

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