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SUMMARY:Syunji Moriya (Osaka Prefecture University)
DTSTART:20200811T041500Z
DTEND:20200811T051500Z
DTSTAMP:20260423T021726Z
UID:operad/5
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/operad/5/">A
  spectral sequence for cohomology of knot spaces</a>\nby Syunji Moriya (Os
 aka Prefecture University) as part of operad pop-up\n\n\nAbstract\nThis ta
 lk is based on the preprint arXiv:2003.03815.\nLet $Emb(S^1\,M)$ be the sp
 ace of embeddings from $S^1$ to a closed manifold $M$ (space of knots in $
 M$). Recently\, this space is studied by Arone-Szymik\, Budney-Gabai\, and
  Kupers\, using Goodwillie-Weiss embedding calculus. In this talk\, we int
 roduce a spectral sequence for cohomology of $Emb(S^1\,M)$ whose $E_2$-ter
 m has an algebraic presentation\, using Sinha's cosimplicial model which i
 s derived from the calculus. This converges to the correct target if $M$ i
 s  simply connected and of dimension $\\geq 4$ for general coefficient rin
 g. Using this\, we see a computation  of $H^*(Emb(S^1\,S^k\\times S^l))$ i
 n low degrees under some assumption on $k\,l$ and an isomorphism \n $\\pi_
 1(Emb(S^1\,M))\\cong H_2(M\,\\mathbb{Z})$ for some simply connected $4$-di
 mensional  $M$. \n\nOur main idea of the construction is to replace   conf
 iguration spaces in the cosimplicial model with fat diagonals via Poincar
 é Lefschetz duality. To do this\, we use a notion of a (co)module over an
  operad. A somewhat curious point is that we need spectra (in stable homot
 opy) even though our concern is singular cohomology.\n
LOCATION:https://researchseminars.org/talk/operad/5/
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