arXiv subjects

Jeffrey Giansiracusa

Publications and source records attributed to Jeffrey Giansiracusa.

31 records · Page 2Linked to original sources

arXiv2017

The diffeomorphism group of a K3 surface and Nielsen realization

The Nielsen Realization problem asks when the group homomorphism from Diff(M) to pi_0 Diff(M) admits a section. For M a closed surface, Kerckhoff proved that a section exists over any finite subgroup, but Morita proved that if the genus is large enough then no section exists over the entire mapping class group. We prove the first nonexistence theorem of this type in dimension 4: if M is a smooth closed oriented 4-manifold which contains a K3 surface as a connected summand then no section exists over the whole of the mapping class group. This is done by showing that certain obstructions lying in the rational cohomology of B(pi_0 Diff(M)) are nonzero. We detect these classes by showing that they are nonzero when pulled back to the moduli space of Einstein metrics on a K3 surface.

math.GT↗

arXiv2017

A Grassmann algebra for matroids

We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plucker vector, which we view as a tensor, and a tropical linear space is recovered from its Plucker vector as the kernel of the corresponding wedge multiplication map. We prove that an arbitrary d-tensor satisfies the tropical Plucker relations (valuated exchange axiom) if and only if the d-th wedge power of the kernel of wedge-multiplication is free of rank one. This provides a new cryptomorphism for valuated matroids, including ordinary matroids as a special case.

math.AG↗

arXiv2016

On the relation between hyperrings and fuzzy rings

We construct a full embedding of the category of hyperfields into Dress's category of fuzzy rings and explicitly characterize the essential image --- it fails to be essentially surjective in a very minor way. This embedding provides an identification of Baker's theory of matroids over hyperfields with Dress's theory of matroids over fuzzy rings (provided one restricts to those fuzzy rings in the essential image). The embedding functor extends from hyperfields to hyperrings, and we study this extension in detail. We also analyze the relation between hyperfields and Baker's partial demifields.

math.CO↗

arXiv2016

On the Hochschild homology of involutive algebras

We study the homological algebra of bimodules over involutive associative algebras. We show that Braun's definition of involutive Hochschild cohomology in terms of the complex of involution-preserving derivations is indeed computing a derived functor: the Z/2-invariants intersected with the center. We then introduce the corresponding involutive Hochschild homology theory and describe it as the derived functor of the pushout of Z/2-coinvariants and abelianization.

math.AT↗

arXiv2016

Equations of tropical varieties

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of $\mathbb{T}$-points this reduces to Kajiwara-Payne's extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of $\mathbb{T}$-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.

math.AG↗

arXiv2011

Invariance properties of Miller-Morita-Mumford characteristic numbers of fibre bundles

Characteristic classes of fibre bundles $E^{d+n}\to B^n$ in the category of closed oriented manifolds give rise to characteristic numbers by integrating the classes over the base. Church, Farb and Thibault [CFT] raised the question of which generalised Miller-Morita-Mumford classes have the property that the associated characteristic number is independent of the fibering and depends only on the cobordism class of the total space $E$. Here we determine a complete answer to this question in both the oriented category and the stably almost complex category. An MMM class has this property if and only if it is a fibre integral of a vector bundle characteristic class that satisfies a certain approximate version of the additivity of the Chern character.

math.AT↗

arXiv2011

Vanishing of universal characteristic classes for handlebody groups and boundary bundles

Using certain Thom spectra appearing in the study of cobordism categories, we show that the odd half of the Miller-Morita-Mumford classes on the mappping class group of a surface with negative Euler characteristic vanish in integral cohomology when restricted to the handlebody subgroup. This is a special case of a more general theorem valid in all dimensions: universal characteristic classes made from monomials in the Pontrjagin classes (and even powers of the Euler class) vanish when pulled back from BDiff(\partial W) to BDiff(W).

math.GT↗

arXiv2010

Formality of the framed little 2-discs operad and semidirect products

We prove that the operad of framed little 2-discs is formal. Tamarkin and Kontsevich each proved that the unframed 2-discs operad is formal. The unframed 2-discs is an operad in the category of S^1-spaces, and the framed 2-discs operad can be constructed from the unframed 2-discs by forming the operadic semidirect product with the circle group. The idea of our proof is to show that Kontsevich's chain of quasi-isomorphisms is compatible with the circle actions and so one can essentially take the operadic semidirect product with the homology of S^1 everywhere to obtain a chain of quasi-isomorphisms between the homology and the chains of the framed 2-discs.

math.QA↗

arXiv2010

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.

math.QA↗

arXiv2010

The framed little 2-discs operad and diffeomorphisms of handlebodies

The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-dimensional handlebodies with marked discs on their boundaries. A modification of the argument provides a new and elementary proof of K. Costello's theorem that the derived modular envelope of the associative operad is homotopy equivalent to the ``open string'' modular operad made from moduli spaces of Riemann surfaces with marked intervals on the boundary. Our technique also recovers a theorem of C. Braun that the derived modular envelope of the cyclic operad that describes associative algebras with involution is homotopy equivalent to the modular operad made from moduli spaces of unoriented Klein surfaces with open string gluing.

math.GT↗

arXiv2009

On the homotopy type of the Deligne-Mumford compactification

An old theorem of Charney and Lee says that the classifying space of the category of stable nodal topological surfaces and isotopy classes of degenerations has the same rational homology as the Deligne-Mumford compactification. We give an integral refinement: the classifying space of the Charney-Lee category actually has the same homotopy type as the moduli stack of stable curves, and the etale homotopy type of the moduli stack is equivalent to the profinite completion of the classifying space of the Charney-Lee category.

math.AT↗

arXiv2008

Pontrjagin-Thom maps and the homology of the moduli stack of stable curves

We study the singular homology (with field coefficients) of the moduli stack of stable n-pointed complex curves of genus g (the Deligne-Mumford compactification). Each of its irreducible boundary components determines via the Pontrjagin-Thom construction a map to a certain infinite loop space whose homology is well understood. We show that these maps are surjective on homology in a range of degrees proportional to the genus. This implies the existence of many new torsion classes in the homology of the moduli stack.

math.AT↗

arXiv2007

The stable mapping class group of simply connected 4-manifolds

We consider mapping class groups Γ(M) = pi_0 Diff(M fix \partial M) of smooth compact simply connected oriented 4-manifolds M bounded by a collection of 3-spheres. We show that if M contains CP^2 (with either orientation) as a connected summand then Γ(M) is independent of the number of boundary components. By repackaging classical results of Wall, Kreck and Quinn, we show that the natural homomorphism from the mapping class group to the group of automorphisms of the intersection form becomes an isomorphism after stabilization with respect to connected sum with CP^2 # \bar{CP^2}. We next consider the 3+1 dimensional cobordism 2-category of 3-spheres, 4-manifolds (as above) and enriched with isotopy classes of diffeomorphisms as 2-morphisms. We identify the homotopy type of the classifying space of this category as the Hermitian algebraic K-theory of the integers. We also comment on versions of these results for simply connected spin 4-manifolds. Finally, we observe that a related 4-manifold operad detects infinite loop spaces.

math.GT↗