The Loop Homology Algebra of Spheres and Projective Spaces

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∗(LSn) and H∗(LCPn). Introduction The loop homology of a closed orientable manifold M of degree d is the ordinary homology of the free loop space LM = Map(S,M), with degree shifted by −d, i.e. H∗(LM ;Z) = H∗+d(LM ;Z). In [3], Chas and Sullivan defined a type of intersection product on the chains of LM , yielding an algebra structure on H∗(LM). Roughly, the loop product is defined as follows. Let α : ∆ → LM and β : ∆ → LM be singular simplices in LM . The evaluation at 1 ∈ S ⊂ C defines a map ev : LM → M . Assume that ev ◦ α and ev ◦ β define a map ∆ ×∆ → LM × LM → M ×M that is transverse to the diagonal. At each point (s, t) ∈ ∆ ×∆ where ev ◦α intersects ev ◦ β, one can define a single loop by first traversing the loop α(s) and then traversing the loop β(t). This then defines a chain α ◦ β ∈ Cp+q−d(LM). In [3] Chas and Sullivan showed that this procedure defines a chain map Cp(LM)⊗ Cq(LM) → Cp+q−d(LM) which induces an associative, commutative algebra structure on the loop homology, H∗(LM). Chas and Sullivan also described other structures this pairing induces, such as a Lie algebra structure on the equivariant homology of the loop space. In [2], the first two authors used the Pontryagin Date: February 1, 2008. The first author was partially supported by a grant from the NSF. 1 2 R.L. COHEN, J.D.S JONES, AND J. YAN Thom construction to show that the loop product is realized on the homotopy level on Thom spaces (spectra) of bundles over the loop space. In particular let TM denote the tangent bundle of M , and −TM denotes its inverse as a virtual bundle in K theory. Let M denote the Thom spectrum of this bundle, and LM the Thom spectrum of ev(−TM). Then in [2] it was shown that LM is a homotopy commutative ring spectrum with unit, whose product realizes the Chas Sullivan product in homology, after applying the Thom isomorphism, Hq(LM −TM ) ∼= Hq+d(LM) ∼= Hq(LM). The goal of this paper is to describe a spectral sequence of algebras converging to the loop homology algebra of a manifold, and to use it to compute the loop homology algebra of spheres and projective spaces. More specifically we shall prove the following theorems. Theorem 1. Let M be a closed, oriented, simply connected manifold. There is a second quadrant spectral sequence of algebras {E p,q, d r : p ≤ 0, q ≥ 0} such that (1) E ∗,∗ is an algebra and the differential d r : E ∗,∗ → E r ∗−r,∗+r−1 is a derivation for each r ≥ 1. (2) The spectral sequence converges to the loop homology H∗(LM) as algebras. That is, E ∞ ∗,∗ is the associated graded algebra to a natural filtration of the algebra H∗(LM). (3) For m,n ≥ 0, E −m,n ∼= H(M ;Hn(ΩM)). Here ΩM is the space of base point preserving loops in M . Furthermore the isomorphism E −∗,∗ ∼= H(M ;H∗(ΩM)) is an isomorphism of algebras, where the algebra structure on H(M ;H∗(ΩM)) is given by the cup product on the cohomology of M with coefficients in the Pontrjagin ring H∗(ΩM). (4) The spectral sequence is natural with respect to smooth maps between manifolds. We then use this spectral sequence to do the following calculations. Let Λ[x1, · · · , xn] denote the exterior algebra (over the integers) generated by x1, · · ·xn, and let Z[a1, · · · , am] denote the polynomial algebra generated by a1, · · · , am. Theorem 2. There exist isomorphisms of graded algebras, (1) H∗(LS ) ∼= Λ[a]⊗ Z[t, t] where a ∈ H−1(LS ) and t, t ∈ H0(LS ). (2) For n > 1, H∗(LS ) = 