The Loop Homology Algebra of Spheres and Projective Spaces

By R. Cohen, John D. S. Jones, Junrong Yan. 2002

129 citations recorded by the source corpus.

In [3] Chas and Sullivan defined an intersection product on the homology H * (LM)of the space of smooth loops in a closed, oriented manifold M.In this paper we will use the homotopy theoretic realization of this product described by the first two authors in [2] to construct a second quadrant spectral sequence of algebras converging to the loop homology multiplicatively, when M is simply connected. The E2 term of this spectral sequence is H * (M;H *(ΩM)where the product is given by the cup product on the cohomology of the manifold H * (M)with coefficients in the Pontryagin ring structure on the homology of its based loop space H *(ΩM)We then use this spectral sequence to compute the ring structures of H * (LS n)and H * (L \( {H_*}\left( {L\mathbb{C}{\mathbb{P}^n}} \right). \))

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c106e78a69f4c77607a0c75b540b598186c44624

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Citations in the corpus

Loop Products and Closed Geodesics (2007, methodology): “The full Chas-Sullivan ring H∗(�) was computed by R. Cohen, J. Jones, and J. Yan [ CJY ] for spheres and projective spaces.”
Membrane Topology (2006, background): “Cohen and Jones, see [6], [7], [8] and [9] , showed that the construction of the intersection product for certain infinite dimensional manifolds reduces to the construction of the umkehr map F! : H(N ) −→ H(M), for F a smooth map between infinite dimensional manifolds N and M.”
Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture (2011, methodology): “To continue this map into C∗−k(A|M ) we simply do as in [6] and pick a Thom class τ in C∗(Dν, Sν) which vanishes on simplices contained in Sν and”
Rational string topology (2004, background): “ (an−1,0) n = 1, 1 •an,1 = 0. This shows that H∗(L(CPn);Q) ∼=∧(a,b,t)/(an+1,anb,ant), with |a|= −2, |b|= −1 and |t|= 2n, a= an−1,0, b= bn−1,0, t= an,1. Note that this computation can also be found in [6] as an application of a spectral sequence deﬁned by Cohen, Jones and Yan. 4 The string bracket Fo”
The Hochschild cohomology of a closed manifold (2004, unknown): “Therefore a set of generators is given by the elements c = 1 ⊗ v2 , b = u ⊗ v , a = u ⊗ 1, |a| = −n , |b| = −1 , |c| = 2n − 2 and, HH∗(C ∗(Sn);C ∗(Sn)) ∼= ∧(b) ⊗ k[a, c]/(2ac, a(2), ab) (see also [5]) .”
Generalized string topology operations (2006, background): “Cohen, Jones and Yan proved the following theorem for the case E = LM in [12] and their proof generalizes to the case of fiberwise monoids.”