Corpus ID: 212633886

# Models for knot spaces and Atiyah duality

@article{Moriya2020ModelsFK,
title={Models for knot spaces and Atiyah duality},
author={S. Moriya},
journal={arXiv: Algebraic Topology},
year={2020}
}
• S. Moriya
• Published 2020
• Mathematics
• arXiv: Algebraic Topology
Let $\mathrm{Emb}(S^1,M)$ be the space of smooth embeddings from the circle to a closed manifold $M$ of dimension $\geq 4$. We study a cosimplicial model of $\mathrm{Emb}(S^1,M)$ in stable categories, using a spectral version of Poincare-Lefschetz duality called Atiyah duality. We actually deal with a notion of a comodule instead of the cosimplicial model, and prove a comodule version of the duality. As an application, we introduce a new spectral sequence converging to $H^*(\mathrm{Emb}(S^1,M… Expand 3 Citations #### Figures from this paper On Cohen-Jones isomorphism in string topology The loop product is an operation in string topology. Cohen and Jones gave a homotopy theoretic realization of the loop product as a classical ring spectrum$LM^{-TM}$for a manifold$M$. Using this,Expand Self-Referential Discs and the Light Bulb Lemma We show how self-referential discs in 4-manifolds lead to the construction of pairs of discs with a common geometrically dual sphere which are properly homotopic rel$\partial$and coincide nearExpand Spaces of knotted circles and exotic smooth structures • Mathematics • 2019 Suppose that$N_1$and$N_2$are closed smooth manifolds of dimension$n$that are homeomorphic. We prove that the spaces of smooth knots$Emb(S^1, N_1)$and$Emb(S^1, N_2)$have the same homotopyExpand #### References SHOWING 1-10 OF 47 REFERENCES Multiplicative properties of Atiyah duality Let$M^n$be a closed, connected$n$-manifold. Let$\mtm$denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that$\mtm$is homotopy equivalent to theExpand On Cohen-Jones isomorphism in string topology The loop product is an operation in string topology. Cohen and Jones gave a homotopy theoretic realization of the loop product as a classical ring spectrum$LM^{-TM}$for a manifold$M$. Using this,Expand MODEL CATEGORIES OF DIAGRAM SPECTRA • Mathematics • 2001 Working in the category$\mathcal{T}$of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functors$\mathcal{D}\longrightarrow \mathcal{T}\$ for a suitableExpand
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A homotopy theoretic realization of string topology
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Abstract. Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In [2] Chas and Sullivan defined a product on the homology H*(LM) of degree -d. They thenExpand
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We study a spectral sequence converging to the cohomology of the configuration space of n ordered points in a manifold. A chain complex is constructed with homology equal to the E2 term. If the fieldExpand