Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.Comment: 93 pages; v4: minor changes; to appear in Advances in Mathematic
A basic result in equivariant K-theory, the Atiyah-Segal completion theorem relates the G-equivariant K-theory of a finite G-CW complex to the nonequivariant K-theory of its Borel construction. We prove the analogous result for twisted equivariant K-theory.
The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in $n$. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in $n$. We also construct nontrivial elements in the mod $p$ homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain poorly understood.Comment: Accepted to the Advances in Mathematic
Abstract. Examples of non-trivial higher string topology operations have been regrettably rare in the literature. In this paper, working in the context of string topology of classifying spaces, we provide explicit calculations of a wealth of non-trivial higher string topology operations associated to a number of different Lie groups. As an application of these calculations, we obtain an abundance of interesting homology classes in the twisted homology groups of automorphism groups of free groups, the ordinary homology groups of holomorphs of free groups, and the ordinary homology groups of affine groups over the integers and the field of two elements.
The mod p homology of E∞-spaces is a classical topic in algebraic topology traditionally approached in terms of Dyer-Lashof operations. In this paper, we offer a new perspective on the subject by providing a detailed investigation of an alternative family of homology operations equivalent to, but distinct from, the Dyer-Lashof operations. Among other things, we will relate these operations to the Dyer-Lashof operations, describe the algebra generated by them, and use them to describe the homology of free E∞-spaces. We will also investigate the relationship between the operations arising from the additive and multiplicative E∞-structures on an E∞-ring space. The operations have especially good properties in this context, allowing for a simple and conceptual formulation of "mixed Adem relations" describing how the operations arising from the two different E∞-structures interact. Contents 1. Introduction 1 2. The E-operations 4 3. The algebra of E-operations 11 4. The homology of free E ∞ -spaces 21 5. The coalgebraic perspective 25 6. The stability and transgression properties 33 7. The homology of E ∞ -ring spaces 35 Acknowledgements 46 References 46
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