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Stability phenomena in the topology of moduli spaces. Surveys in Differential Geometry XIV ed
 and ST Yau, International Press (2010
"... The recent proof by Madsen and Weiss of Mumford’s conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spac ..."
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The recent proof by Madsen and Weiss of Mumford’s conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where “stability phenomena ” occur in their topologies. Such stability theorems have been proved in many situations in the history of topology and geometry, and the payoff has often been quite remarkable. In this paper we discuss classical stability theorems such as the Freudenthal suspension theorem, Bott periodicity, and Whitney’s embedding theorems. We then discuss more modern examples such as those involving configuration spaces of points in manifolds, holomorphic curves in complex manifolds, gauge theoretic moduli spaces, the stable topology of general linear groups, and pseudoisotopies of manifolds. We then discuss the stability theorems regarding the moduli spaces of Riemann surfaces: Harer’s stability theorem on the cohomology of moduli space, and the MadsenWeiss theorem, which proves a generalization of Mumford’s conjecture. We also describe Galatius’s recent theorem on the stable cohomology of automorphisms of free groups. We end by speculating on the existence of general conditions in
HOLOMORPHIC K THEORY, ALGEBRAIC COCYCLES, AND LOOP GROUPS
"... Abstract. In this paper we study the “holomorphic Ktheory ” of a projective variety. This K theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory has been introduced in various places such as [12], [9], and a re ..."
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Abstract. In this paper we study the “holomorphic Ktheory ” of a projective variety. This K theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory has been introduced in various places such as [12], [9], and a related theory was considered in [11]. This theory is built out of studying algebraic bundles over a variety up to “algebraic equivalence”. In this paper we will give calculations of this theory for “flag like varieties ” which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors in [6], we will show that there is a rational isomorphism of graded rings between holomorphic K theory and the appropriate “morphic cohomology ” groups, defined in [7] in terms of algebraic cocycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the “symmetrized loop group ” ΩU(n)/Σn where the symmetric group Σn acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K theory by inverting the Bott class, then rationally this is isomorphic to topological K theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K theory, and show that these obstructions vanish for generalized flag manifolds.
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, 812
"... Spaces of algebraic maps from real projective spaces into complex projective spaces ..."
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Spaces of algebraic maps from real projective spaces into complex projective spaces
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, 809
"... Spaces of algebraic and continuous maps between real algebraic varieties ..."
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, 809
"... Spaces of algebraic and continuous maps between real algebraic varieties ..."