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I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be $\small \dim \tilde{H}_t(X; {\mathbb{Z}}_2) = \begin{cases} 1+\sum_{r=k+1}^{2k} \sum_{J= 1}^{2r-3}\binom{2k-r+J}{J}\binom{2r-2k-J-1}{J-1}, &t=2k, k\ge 1, \\ \sum_{r=k+2}^{2k+1} \sum_{J= 1}^{r-k-1}\binom{2k-r+J+1}{J}\binom{2r-2k-J-2}{J-1}, &t=2k+1, k\ge 0. \end{cases} $

I then search the Online Encyclopedia of Integer Sequences and find that this sequence is A052547, except for in dimension 0 (perhaps because I am using reduced homology). From the description of A052547 as the expansion of $(1-x)/(x^3-2x^2-x+1)$, I conjecture that the reduced $\bmod 2$ Poincaré series is $\sum_{t=0}^\infty \dim\tilde{H}_t(X;{\mathbb{Z}}_2) \,x^t = \frac{1-x}{x^3 - 2x^2 - x+1} - 1.$

How do I prove this conjecture?

  • 0
    Have you tried just handing the sum to a CAS? It looks like the kind of thing that Zeilberger's algorithm should handle automatically. – 2017-10-20

0 Answers 0