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Derived string topology and the Eilenberg-Moore spectral sequence

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Abstract

Let M be a simply-connected closed manifold of dimension m. Chas and Sullivan have defined (co)products on the homology of the free loop space H*(LM). Félix and Thomas have extended the loop (co)products to those of simply-connected Gorenstein spaces over a field. We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories.

In Algebraic Topology, one of the most important tools for computing the (co)homology of the space of free loops on a space is the (co)homological Eilenberg-Moore spectral sequence. Consider, over any field, the homological Eilenberg-Moore spectral sequence converging to H *(LM). Our description of the loop product enables one to conclude that this spectral sequence is multiplicative with respect to the Chas-Sullivan loop product and that its E 2-term is the Hochschild cohomology of H*(M). This gives a new method to compute the loop products on H *(LS m) and H *(LℂP r), the free loop space homology of spheres and complex projective spaces.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto, Nagano, 390-8621, Japan

    Katsuhiko Kuribayashi

  2. LAREMA - UMR CNRS 6093, Université dĩAngers, 2 Bd Lavoisier, 49045, Angers, France

    Luc Menichi

  3. Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto, Nagano, 390-8621, Japan

    Takahito Naito

Authors
  1. Katsuhiko Kuribayashi
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  2. Luc Menichi
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  3. Takahito Naito
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Corresponding author

Correspondence to Katsuhiko Kuribayashi.

Additional information

The first author was partially supported by a Grant-in-Aid for Scientific Research (B) 25287008 from Japan Society for the Promotion of Science.

The third author was supported by JSPS Fellowships for Young Scientists No. 24-10827.

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Kuribayashi, K., Menichi, L. & Naito, T. Derived string topology and the Eilenberg-Moore spectral sequence. Isr. J. Math. 209, 745–802 (2015). https://doi.org/10.1007/s11856-015-1236-y

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  • DOI: https://doi.org/10.1007/s11856-015-1236-y

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