Rational cohomology of configuration spaces of surfaces

Bödigheimer, Carl-Friedrich; Cohen, Fred

doi:10.1007/bfb0083031

Cited by 21 publications

(23 citation statements)

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“…In this section, we compute the Betti numbers of

B_{k} false(normalΣ false)

, where

normalΣ

is either nonorientable or both orientable and open. The bulk of the former computation was carried out in and the bulk of the latter in ; nevertheless, we include them here for the sake of completeness and as a warmup for the more involved computations to come.…”

Section: Nonorientable and Open Surfacesmentioning

confidence: 99%

“…The Lie algebra homology of

frakturh

was computed in dual form in . Theorem

{prefixdim}_{i, k} false(H (C E false(frakturh false)) = \{\begin{matrix} left (\frac{0pt}{2 g}) - (\frac{0pt}{2 g}) & left 0 ⩽ i ⩽ g, i = k \\ left (\frac{0pt}{2 g}) - (\frac{0pt}{2 g}) & left g + 1 ⩽ i ⩽ 2 g + 1, i = k - 1 \\ left 0 & left else. \end{matrix}

…”

Section: Nonorientable and Open Surfacesmentioning

confidence: 99%

“…All of the calculations below are performed with a differential, also called D, that omits the sign (−1) |α|+|β|+1 . This omission is justified by a linear change of variables; indeed, multiplication by −1 in the even part of H − * c (Σ; Q) [3] eliminates the sign.…”

Section: Configuration Spaces and Lie Algebrasmentioning

confidence: 99%

“…Some of these computations are known. The case of the plane is treated by Arnold in , the case of a once‐punctured orientable surface by Bödigheimer–Cohen in , the case of the sphere by Salvatore in (see also ), the case of the real projective plane by Wang in , and the case of a general closed nonorientable surface by the second author in . In addition, during the writing of this work, the authors learned of two independent computations in the case of the torus, due to Maguire and Schiessl , respectively.…”

Section: Introductionmentioning

confidence: 99%

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Betti numbers of configuration spaces of surfaces

Drummond-Cole

Knudsen

2017

J. London Math. Soc.

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“…In this section, we compute the Betti numbers of

B_{k} false(normalΣ false)

, where

normalΣ

Section: Nonorientable and Open Surfacesmentioning

confidence: 99%

“…The Lie algebra homology of

frakturh

was computed in dual form in . Theorem

{prefixdim}_{i, k} false(H (C E false(frakturh false)) = \{\begin{matrix} left (\frac{0pt}{2 g}) - (\frac{0pt}{2 g}) & left 0 ⩽ i ⩽ g, i = k \\ left (\frac{0pt}{2 g}) - (\frac{0pt}{2 g}) & left g + 1 ⩽ i ⩽ 2 g + 1, i = k - 1 \\ left 0 & left else. \end{matrix}

…”

Section: Nonorientable and Open Surfacesmentioning

confidence: 99%

Section: Configuration Spaces and Lie Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Betti numbers of configuration spaces of surfaces

Drummond-Cole

Knudsen

2017

J. London Math. Soc.

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“…The rational calculations were done previously by Cohen-Taylor in [10] and the special case where X is a sphere was done by BodigheimerCohen-Taylor in [5]. Other cases can be found in [4,8,9,11,12,13,20]. The homology with coefficients in a field of characteristic p = 2 of the configuration spaces C(M, S n ) such that dimM + n is even still remains open.…”

Section: Introductionmentioning

confidence: 99%