# 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} }

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

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#### References

SHOWING 1-10 OF 47 REFERENCES

Multiplicative properties of Atiyah duality

- Mathematics
- 2004

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 the… Expand

On Cohen-Jones isomorphism in string topology

- Mathematics
- 2020

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 suitable… Expand

The Lambrechts–Stanley model of configuration spaces

- Mathematics
- 2016

We prove the validity over $${\mathbb {R}}$$R of a commutative differential graded algebra model of configuration spaces for simply connected closed smooth manifolds, answering a conjecture of… Expand

A family of embedding spaces

- Mathematics
- 2006

Let Emb(S^j,S^n) denote the space of C^infty-smooth embeddings of the j-sphere in the n-sphere. This paper considers homotopy-theoretic properties of the family of spaces Emb(S^j,S^n) for n >= j > 0.… Expand

A homotopy theoretic realization of string topology

- Mathematics
- 2001

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 then… Expand

The cohomology of certain function spaces

- Mathematics
- 1991

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 field… Expand

Operads and knot spaces

- Mathematics
- 2004

Let Em denote the space of embeddings of the interval I = [?1,1] in the cube Im with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy… Expand

Galois symmetries of knot spaces

- Mathematics
- Compositio Mathematica
- 2021

We exploit the Galois symmetries of the little disks operads to show that many differentials in the Goodwillie–Weiss spectral sequences approximating the homology and homotopy of knot spaces vanish… Expand

Surgery on Simply-Connected Manifolds

- Mathematics
- 1972

I. Poincare Duality.- 1. Slant Operations, Cup and Cap Products.- 2. Poincare Duality.- 3. Poincare Pairs and Triads Sums of Poincare Pairs and Maps.- 4. The Spivak Normal Fibre Space.- II. The Main… Expand