133 research outputs found
Derived Algebraic Geometry
This text is a survey of derived algebraic geometry. It covers a variety of
general notions and results from the subject with a view on the recent
developments at the interface with deformation quantization.Comment: Final version. To appear in EMS Surveys in Mathematical Science
Subdivisional spaces and graph braid groups
We study the problem of computing the homology of the configuration spaces of
a finite cell complex . We proceed by viewing , together with its
subdivisions, as a subdivisional space--a kind of diagram object in a category
of cell complexes. After developing a version of Morse theory for subdivisional
spaces, we decompose and show that the homology of the configuration spaces
of is computed by the derived tensor product of the Morse complexes of the
pieces of the decomposition, an analogue of the monoidal excision property of
factorization homology.
Applying this theory to the configuration spaces of a graph, we recover a
cellular chain model due to \'{S}wi\k{a}tkowski. Our method of deriving this
model enhances it with various convenient functorialities, exact sequences, and
module structures, which we exploit in numerous computations, old and new.Comment: 71 pages, 15 figures. Typo fixed. May differ slightly from version
published in Documenta Mathematic
The refined transfer, bundle structures and algebraic K-theory
We give new homotopy theoretic criteria for deciding when a fibration with
homotopy finite fibers admits a reduction to a fiber bundle with compact
topological manifold fibers. The criteria lead to a new and unexpected result
about homeomorphism groups of manifolds. A tool used in the proof is a
surjective splitting of the assembly map for Waldhausen's functor A(X).
We also give concrete examples of fibrations having a reduction to a fiber
bundle with compact topological manifold fibers but which fail to admit a
compact fiber smoothing. The examples are detected by algebraic K-theory
invariants.
We consider a refinement of the Becker-Gottlieb transfer. We show that a
version of the axioms described by Becker and Schultz uniquely determines the
refined transfer for the class of fibrations admitting a reduction to a fiber
bundle with compact topological manifold fibers.
In an appendix, we sketch a theory of characteristic classes for fibrations.
The classes are primary obstructions to finding a compact fiber smoothing.Comment: This version contains mostly minor revision
Higher Segal spaces I
This is the first paper in a series on new higher categorical structures
called higher Segal spaces. For every d > 0, we introduce the notion of a
d-Segal space which is a simplicial space satisfying locality conditions
related to triangulations of cyclic polytopes of dimension d. In the case d=1,
we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal
spaces. The starting point of the theory is the observation that Hall algebras,
as previously studied, are only the shadow of a much richer structure governed
by a system of higher coherences captured in the datum of a 2-Segal space. This
2-Segal space is given by Waldhausen's S-construction, a simplicial space
familiar in algebraic K-theory. Other examples of 2-Segal spaces arise
naturally in classical topics such as Hecke algebras, cyclic bar constructions,
configuration spaces of flags, solutions of the pentagon equation, and mapping
class groups.Comment: 221 page
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