Wikipedia:Reference desk/Archives/Mathematics/2011 March 1

= March 1 =

Value of delta
Find a value of delta in terms of epsilon such that $$|x-2|< \delta \Rightarrow \left|\frac{1}{x}-\frac{1}{2}\right|< \epsilon $$

$$\left|\frac{1}{x}-\frac{1}{2}\right|$$

$$=\left|\frac{2-x}{2x}\right|$$

$$\le \left|\frac{\delta}{2x}\right|< \epsilon$$

but there is still a pesky x in the denominator. Widener (talk) 11:36, 1 March 2011 (UTC)
 * Hint: $$\delta = \min(1,2\epsilon)$$ works. First prove that this is indeed the case; then figure out how you could have found it without knowing it in advance. -- Meni Rosenfeld (talk) 12:13, 1 March 2011 (UTC)


 * The answer can be easily seen on a graph, but if you like to solve your problem by algebraic means I'd suggest to plug a new symbol into your expressions. Define $$ y = x-2 $$ which implies $$x = y+2$$ and $$ \frac 1 x - \frac 1 2 = {-y \over 2(y+2)}$$. Then the question reads: 'Which y satisfy the inequality $$\left| {-y \over 2(y+2)} \right| < \epsilon$$? What is their absolute values' upper bound δ in terms of ε?' That can be transformed into a pair of linear inequalities (&#124;m&#124; &lt; q ⇔ &minus;q &lt; m &lt; q) — but don't forget to consider intervals, where denominator is positive and where it's negative, because 'less than' becomes 'greater than' on multiplying both sides by negative value. --CiaPan (talk) 06:41, 2 March 2011 (UTC)

Self-homeomorphisms and fundamental groups
Let X be a topological space. When $$f:X \to X$$ is a homeomorphism, then $$f_*$$, the induced homomorphism on the fundamental group, is an automorphism. Is the converse true? I.e. if $$f_*$$ is an automorphism, then is f homotopic to a homeomorphism? I know it's true for some nice spaces (tori). Is this true for all spaces? For compact manifolds? Any reference discussing this would be appreciated. Staecker (talk) 17:27, 1 March 2011 (UTC)
 * No. Something homotopic to a homeomorphism has to induce an isomorphism on homology, cohomology, higher homotopy and so on, while inducing an isomorphism on the fundamental group is a pretty weak condition, especially if the fundamental group is trivial. For example, any self-map of S2 with degree not 1 or -1 (such as a constant map, say) is a counterexample. Algebraist 18:15, 1 March 2011 (UTC)
 * Yes I should have thought of that- thanks! Staecker (talk) 18:34, 1 March 2011 (UTC)
 * Also, the fundamental group, and the other invariants Algebraist mentioned, are "homotopy invariant" (see homotopy), so any homotopy equivalence will induce an isomorphism on these. And being a homotopy equivalence is a lot weaker than being a homeomorphism. Aenar (talk) 18:57, 1 March 2011 (UTC)
 * But how does being a homotopy equivalence from X to X compare with being homotopic to a homeomorphism? Algebraist 19:01, 1 March 2011 (UTC)
 * Yeah, sorry, I just realized I hadn't read the question right, and that it is actually a good (hard?) question (especially if reformulated as "for which spaces is any homotopy automorphism homotopic to a homeomorphism"). Aenar (talk) 19:06, 1 March 2011 (UTC)
 * All I've discovered so far is that it seems to be a well-studied question. For example, the Borel conjecture is that the two are the same for aspherical closed manifolds. Whitehead's theorem already implies that any self-map of such a manifold that induces an isomorphism on the fundamental group is a homotopy equivalence, so that (conjecturally) gives a larger class of nice spaces than just tori. Algebraist 19:15, 1 March 2011 (UTC)
 * Interesting. Aenar (talk) 19:28, 1 March 2011 (UTC)
 * Yes very interesting- I hadn't heard of the Borel conjecture. Thanks a lot- Staecker (talk) 00:00, 2 March 2011 (UTC)