The conjecture on the stable gap for sequential topological complexity of real projective spaces

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Assume m≡2e−1(mod2e+1)m\equiv2^e-1\pmod{2^{e+1}} with e≥0e\geq0, so ee is the length of the block of consecutive ones ending the binary expansion of mm. Let G(m,s)=sm−zcl⁡s(RP⁡m)G(m,s)=sm-\operatorname{zcl}_s(\operatorname{\mathbb{R}P}^m), and let G(m)G(m) be the stable value of the non-increasing sequence G(m,2)≥G(m,3)≥⋯≥0G(m,2)\geq G(m,3)\geq\cdots\geq0. Stable-gap conjecture. In Theorem 3.3 of the source, the equality G(m)=2e−1G(m)=2^e-1 holds without restriction on ee. This gives the asserted exact stable value for the gap between the dimensional upper bound and the zero-divisor-cup-length lower bound for sequential topological complexity. The source states that this conjecture has recently been proved, so the claim is solved.

References

Primary source

Jesus Gonzalez, Darwin Gutierrez and Adriana Lara, “Projective product coverings and sequential motion planning algorithms in real projective spaces”, arXiv:1605.07966 (2016).

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