The cohomology algebra of certain free loop spaces
نویسنده
چکیده
Let X be a simply connected space and LX the space of free loops on X. We determine the mod p cohomology algebra of LX when the mod p cohomology of X is generated by one element or is an exterior algebra on two generators. We also provide lower bounds on the dimensions of the Hodge decomposition factors of the rational cohomology of LX when the rational cohomology of X is a graded complete intersection algebra. The key to both of these results is the identification of an important subalgebra of the Hochschild homology of a graded complete intersection algebra over a field. 0. Introduction. Let p be a prime number or zero, X a simply connected space and LX the space of free loops on X. In this paper, Z/p means the rational number field Q if p = 0. In order to calculate the mod p cohomology H∗(LX;Z/p) from H∗(X;Z/p), one may use the Eilenberg–Moore spectral sequence ([5], [13], [15]) for the fiber square F(X): LX → X ↓ ↓ ∆ X → ∆ X ×X where ∆ is the diagonal map. In the procedure, the Hochschild homology HH∗(H∗(X;Z/p), 0) of the commutative differential graded algebra (DGA) (H∗(X;Z/p), 0) with the trivial differential appears. In fact, the E2-term of the spectral sequence for F(X) is isomorphic to TorH∗(X;Z/p)⊗H∗(X;Z/p)(H∗(X;Z/p),H∗(X;Z/p)), that is, HH∗(H∗(X;Z/p), 0) with an appropriate bigrading as a bigraded algebra. If H∗(X;Z/p) is a graded complete intersection algebra (GCIalgebra), there is a DGA ([15], [11]) whose cohomology is isomorphic to the E2-term. 1991 Mathematics Subject Classification: 55N35, 55P35, 55P62, 55T20.
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