Paolo Salvatore

Generalized string topology operations

We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and the intersection product on the base, makes sense on the total space homology of any fiberwise monoid E over a closed oriented manifold M. More generally, the Thom spectrum E-TM is a ring spectrum. Similarly, a fiberwise module over E defines a module over E-TM Fiberwise monoids include adjoint bundles of principal bundles, and the construction is natural with respect to maps of principal bundles. This naturality implies homotopy invariance of the algebra structure on H-*(LM) arising from the loop product. If M = BG is the infinite-dimensional classifying space of a compact Lie group, then we get a well-defined pro-ring spectrum, which we define to be the string topology of BG. If E has a fiberwise action of the little n-cubes operad then E-TM is an E-n-ring spectrum. This gives homology operations combining Dyer-Lashof operations on the fiber and the Poincare duals of Steenrod operations on the base. We give several examples where the new operations give homological insight, borrowed from knot theory, complex geometry, gauge theory, and homotopy theory.

Configuration spaces with summable labels

An n-monoid is the appropriate extension of an A ∞-space for the theory of n-fold loop spaces. We define spaces of configurations on n-manifolds with summable labels in partial n-monoids. In particular we obtain an n-fold delooping machinery, that extends the construction of the classifying space by Stasheff. Our configuration spaces cover also symmetric products, spaces of rational curves and spaces of labelled subsets. A configuration space with connected space of labels has the homotopy type of the space of sections of a certain bundle. This extends and unifies results by Bodigheimer, Guest, Kallel and May.

Knots, operads, and double loop spaces

We show that the space of long knots in an Euclidean space of dimension larger than three is a double loop space, proving a conjecture by Sinha. We also construct a double loop space structure on framed long knots, and show that the map forgetting the framing is not a double loop map in odd dimension. However, there is always such a map in the reverse direction expressing the double loop space of framed long knots as a semidirect product. A similar compatible decomposition holds for the homotopy fiber of the inclusion of long knots into immersions. We also show via string topology that the space of closed knots in a sphere, suitably desuspended, admits an action of the little 2-discs operad in the category of spectra. A fundamental tool is the McClure-Smith cosimplicial machinery, that produces double loop spaces out of topological operads with multiplication.

Rational maps and string topology

We apply a version of the Chas-Sullivan-Cohen-Jones product on the higher loop homology of a manifold in order to compute the homology of the spaces of continuous and holomorphic maps of the Riemann sphere into a complex projective space. This product makes sense on the homology of maps from a co-H space to a manifold, and comes from a ring spectrum. We also build a holomorphic version of the product for maps of the Riemann sphere into homogeneous spaces. In the continuous case we define a related module structure on the homology of maps from a mapping cone into a manifold, and then describe a spectral sequence that can compute it. As a consequence we deduce a periodicity and dichotomy theorem when the source is a compact Riemann surface and the target is a complex projective space.

Rational homotopy nilpotency of self-equivalences

Abstract We study the homotopy nilpotency, after rationalization, of some spaces of self-homotopy equivalences of a finite, simply connected CW-complex.

The mod-2 cohomology rings of symmetric groups

On the cohomology of BS• the second product · is cup product, which is zero for classes supported on disjoint components. The first product ⊙ is the relatively new transfer product first studied by Strickland and Turner [21], (see Definition 3.1). It is akin to the “induction product” in the representation theory of symmetric groups, which dates back to Young and has been in standard use [9, 22]. The coproduct ∆ on cohomology is dual to the standard Pontrjagin product on the homology of BS•. This Hopf ring structure was used by Strickland in [20] to calculate the Morava E-theory of symmetric groups. Though Hopf rings were introduced by Milgram to study the homology of the sphere spectrum [11] and thus of symmetric groups [4], the Hopf ring structure we study does not fit into the standard framework. In particular it exists in cohomology rather than homology. See [21] for a lucid, complete discussion of the relationships between all of these structures. But like in calculations such as that of Ravenel and Wilson [17], we find this Hopf ring presentation to be quite efficient, given by a simple list of generators and relations.

Cyclic operad formality for compactified moduli spaces of genus zero surfaces

The framed little 2-discs operad is homotopy equivalent to the Kimura-Stasheff-Voronov cyclic operad of moduli spaces of genus zero stable curves with tangent rays at the marked points and nodes. We show that this cyclic operad is formal, meaning that its chains and its homology (the Batalin-Vilkovisky operad) are quasi-isomorphic cyclic operads. To prove this we introduce a new complex of graphs in which the differential is a combination of edge deletion and contraction, and we show that this complex resolves BV as a cyclic operad.

The space of linear maps into a Grassmann manifold

Abstract. We show that the space of all holomorphic maps of degree one from the Riemann sphere into a Grassmann manifold is a sphere bundle over a flag manifold. Using the notions of “kernel” and “span” of a map, we completely identify the space of unparameterized maps as well. The illustrative case of maps into the quadric Grassmann manifold is discussed in detail and the homology of the corresponding spaces is computed.