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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeNov 25th 2019
• (edited Nov 25th 2019)

starting something, in order to record Theorem 4.2 in

But HELP: The article states this without proof. I understand how the discussion here and in the other articles now cited in the entry goes towards the proof, but I’d really like to see the actual proof written out. Eventually.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeNov 26th 2019
• (edited Nov 26th 2019)

added statement of the stronger version of the theorem, for horizontal chord diagrams (Theorem 4.1 in Kohno 02 )

$H \mathbb{Z}^\bullet \big( \Omega \underset{ {}^{\{1,\cdots,n\}} }{Conf} (\mathbb{R}^3) \big) \;\simeq\; (\mathcal{W}^{pb}_n)^\bullet \;\simeq\; Gr^\bullet( \mathcal{V}^{pb}_n ) \,.$
• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 17th 2019
• (edited Dec 17th 2019)

I have expanded the statement, by

first stating it in the form

$H_\bullet \big( \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^D) \big) \;\simeq\; \mathcal{U} \big( \mathcal{L}_n(D) \big) \,.$

then observing the isomorphism

$\big(\mathcal{A}_n^{pb}, \circ\big) \;\simeq\; \mathcal{U}(\mathcal{L}_n(D)) \,.$

hence the alternative form

$H_\bullet \big( \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^D) \big) \;\simeq\; \big(\mathcal{A}_n^{pb}, \circ\big)$

from which then the final result

$H^\bullet \big( \Omega \underset{ {}^{\{1,\cdots,n\}} }{Conf} (\mathbb{R}^3) \big) \;\simeq\; (\mathcal{W}_n^{pb})^\bullet \;\simeq\; Gr^\bullet( \mathcal{V}_{pb} )$

follows.

The attribution of these stages is dizzying, with Cohen-Gitler and Kohno seemingly citing each other and their own formerly upcoming preprints back and forth, without, it seems, ever dwelling much on spelling out the full proof (?). Luckily there is also Fadell-Husseini 01, Theorem 2.2 for the first stage, and from there the rest is immediate.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJan 14th 2020

added a remark to make explicit the Hopf algebra structure on both sides