Moduli spaces of manifolds: a user's guide

Abstract

We survey recent work on moduli spaces of manifolds with an emphasis on the role played by (stable and unstable) homotopy theory. The theory is illustrated with several worked examples.2010 Mathematics Subject Classification. 57R90, 57R15, 57R56, 55P47. 2.2.Classifying spaces. The natural equivalence relation between the bundles considered above is concordance, which we recall.Definition 2.2. Let π 0 : E 0 → X and π 1 : E 1 → X be smooth bundles with Θ-structures ρ 0 : Fr(T π0 E 0 ) → Θ and ρ 1 : Fr(T π1 E 1 ) → Θ.(i) An isomorphism between (π 0 , ρ 0 ) and (π 1 , ρ 1 ) is a diffeomorphism φ : E 0 → E 1 over X, such that the induced map Fr(D π φ) : Fr(T π0 E 0 ) → Fr(T π1 E 1 ) is over Θ. (ii) A concordance between (π 0 , ρ 0 ) and (π 1 , ρ 1 ) is a smooth fibre bundle π :E → R × X with Θ-structure ρ : Fr(T π E) → Θ, together with isomorphisms from (π 0 , ρ 0 ) and (π 1 , ρ 1 ) to the pullbacks of (π, ρ) along the two embeddingsPulling back is functorial up to isomorphism and preserves being concordant.Definition 2.3. For a smooth manifold X without boundary, let F Θ [X] denote the set of concordance classes of pairs (π, ρ) of a smooth fibre bundle π : E → X with Θ-structure ρ : Fr(T π E) → Θ. (Note that X is fixed, but E is allowed to vary.)Theorem 2.4. The functor X → F Θ [X] is representable in the (weak) sense that there exists a topological space M Θ and a natural bijection 1)where the codomain denotes homotopy classes of continuous maps.

Mapping class groups of highly connected $$(4k+2)$$-manifolds

“…Our main result Theorem B has been used in [32] in conjunction with Galatius-Randal-Williams' work on moduli spaces of manifolds [22] to compute the second stable homology of the theta-subgroup of Sp 2g (Z) (see Sect. 1.2), or equivalently, the second quadratic symplectic algebraic K -theory group of the integers KSp q Outline Section 1 serves to recall foundational material on diffeomorphism groups and their classifying spaces, as well as to introduce different variants of the extensions (1) and (2) and to establish some of their basic properties.…”

“…The description of n g up to these two extension problems has found a variety of applications [2,[5][6][7]18,23,29,33,38,39], especially in relation to the study of moduli spaces of manifolds [22]. The remaining extensions (1) and (2) have been studied more closely for particular values of g and n [15,19,21,36,37,48] but are generally not well-understood (see e.g.…”

The equivariant cobordism category

“…A main application of cobordism categories in the non‐equivariant case is to moduli spaces of manifolds, see the recent survey [10] and the references therein. Let us end this section with some preliminary remarks about a possible application of Theorems 1.1 and 6.4 to moduli spaces of equivariant manifolds, focusing on closed manifolds for simplicity.…”

The classifying space of the one‐dimensional bordism category and a cobordism model for TC of spaces

“…Although we introduce all the necessary tools, we can only do so concisely. The interested reader is referred to [14] for a discussion of moduli spaces for manifolds, and to [6] for an introduction to the world of non‐unital topological categories and semisimplicial spaces. Remark The specific model for the cobordism category that we are using is essentially that of [10].…”

“…One can think of as a concrete topological model for the ‘moduli space of ‐structured closed ‐dimensional manifolds’. In [14, Section 2.2] this space is denoted . The moduli space decomposes as a disjoint union over diffeomorphism types: Fact There is a weak equivalence where runs over a set of representatives of diffeomorphism classes of closed ‐dimensional manifolds.…”