Loop Homology as Fibrewise Homology

Abstract

The loop homology ring of an oriented closed manifold, defined by Chas and Sullivan, is interpreted as a fibrewise homology Pontrjagin ring. The basic structure, particularly the commutativity of the loop multiplication and the homotopy invariance, is explained from the viewpoint of the fibrewise theory, and the definition is extended to arbitrary compact manifolds.

The homotopy invariance of the string topology loop product and string bracket

“…Remark. After the results of this paper were announced, two independent proofs of the homotopy invariance of the loop homology product were found by by Crabb [12], and by Gruher and Salvatore [14].…”

Umkehr maps

“…As was seen in [10] and [4], the ring structure coming from the fiberwise monoid corresponds to the ring spectrum structure on LN −τ N described in [3], and thus reflects the string topology loop product.…”

On the invariance of the string topology coproduct

“…The map ĥ and Diagram (8). We start by defining the map ĥ of Diagram (8). Recall from Proposition 1.4 the diffeomorphism h :…”

“…While the above products and coproducts are all induced by chain maps, we will only consider in the present paper the induced maps in homology/cohomology. It has been shown by several authors that the Chas-Sullivan product is homotopy invariant, in the sense that a homotopy equivalence f : M 1 → M 2 induces an isomorphism Λf : H * (ΛM 1 ) → H * (ΛM 2 ) of algebras with respect to the Chas-Sullivan product (see [7, Thm 1], [12,Prop 23], [8,Thm 3.7]). This result left open the question whether a more complex operation like the Goresky-Hingston coproduct would, or would not, likewise be homotopy invariant.…”