Timeline for Minimal number of cells of a CW complex (up to homotopy)
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 25, 2012 at 6:56 | comment | added | Ryan Budney | The class of spaces where the Q2 formula gives the minimal number of cells (up to a homotopy-equivalence) is fairly narrow. If $H_2$ is trivial and $H_1$ torsion free, that formula says there's no 2-cells. So $\pi_1 X$ has to be a free group. So for a homology sphere to be in your class, it would have to be a homotopy sphere. That's very restrictive. | |
| Oct 25, 2012 at 1:56 | comment | added | Patricia Hersh | @Ryan: is it obvious when the Whitehead group of $\pi_1 X$ is trivial that indeed such an $X'$ exists? Thanks for your help understanding this! | |
| Oct 25, 2012 at 0:42 | history | edited | Ralph | CC BY-SA 3.0 |
typo
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| Oct 25, 2012 at 0:35 | history | edited | Ralph | CC BY-SA 3.0 |
reformulated Q3
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| Oct 25, 2012 at 0:22 | comment | added | Ralph | Thanks for the explanation. However, Q2 doesn't ask for particular counterexamples but for interesting classes of spaces where $(\ast)$ gives the minimal number of cells. | |
| Oct 25, 2012 at 0:03 | comment | added | Ryan Budney | The complement of a knot in the 3-sphere has the same homology as $S^1$. For example, the degree of the Alexander polynomial is the rank of $H_1$ of the universal abelian cover of the knot complement (with rational coefficients). The number of 1-cells must therefore be at least as large as the degree of the Alexander polynomial. | |
| Oct 24, 2012 at 23:54 | comment | added | Ralph | @Ryan: Thanks for your comment and answer. Q2: I thought spaces with abelian fundamental group or classifying spaces might be candidates. I don't understand why your argument ("think of knot complements") completely sorts out these classes. Q3: Can you give some more details on how the improved bound looks like. | |
| Oct 24, 2012 at 23:33 | answer | added | Ryan Budney | timeline score: 5 | |
| Oct 24, 2012 at 23:23 | comment | added | Ryan Budney | The tool to find minimal CW-decompositions of a space (up to homotopy-equivalence) is called Whitehead's Theorem. It's covered in Hatcher's textbook, at the start of Chapter 4.1. So Q1 has an affirmative answer. Q2 has a negative answer (think of knot complements). Q3 yes, of course. Think about the homology of covering spaces as modules over the group of covering transformations, for example | |
| Oct 24, 2012 at 23:13 | history | asked | Ralph | CC BY-SA 3.0 |