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Tadeusz Januszkiewicz

Publications and source records attributed to Tadeusz Januszkiewicz.

15 recordsLinked to original sources
arXiv2024

Tautological characteristic classes II: the Witt class

Let $K$ be an arbitrary infinite field. The cohomology group $H^2(SL(2,K), H_2\,SL(2,K))$ contains the class of the universal central extension. When studying representations of fundamental groups of surfaces in $SL(2,K)$ it is useful to have classes stable under deformations (Fenchel--Nielsen twists) of representations. We identify the maximal quotient of the universal class which is stable under twists as the Witt class of Nekovar. The Milnor--Wood inequality asserts that an $SL(2,{\bf R})$-bundle over a surface of genus $g$ admits a flat structure if and only if its Euler number is $\leq (g-1)$. We establish an analog of this inequality, and a saturation result for the Witt class. The result is sharp for the field of rationals, but not sharp in general.

math.KT↗
arXiv2023

Tautological characteristic classes I

We discuss the formalism of tautological characteristic classes of flat bundles. Applied to $PSL(2,K)$ it yields the Witt class of Nekovar. Applied to $PGL_+(2n,K)$, the general linear groups with positive determinant over an arbitrary ordered field, it yields (a generalization of) the Euler class.

math.AG↗
arXiv2016

Constructions of k-regular maps using finite local schemes

A continuous map from R^m to R^N or from C^m to C^N is called k-regular if the images of any $k$ points are linearly independent. Given integers m and k a problem going back to Chebyshev and Borsuk is to determine the minimal value of N for which such maps exist. The methods of algebraic topology provide lower bounds for N, however there are very few results on the existence of such maps for particular values m and k. Using the methods of algebraic geometry we construct k-regular maps. We relate the upper bounds on N with the dimension of the locus of certain Gorenstein schemes in the punctual Hilbert scheme. The computations of the dimension of this family is explicit for k<10, and we provide explicit examples for k<6. We also provide upper bounds for arbitrary m and k.

math.DG↗
arXiv2010

Cohomology of hyperplane complements with group ring coefficients

We compute the cohomology with group ring coefficients of the complement of a finite collection of affine hyperplanes in a finite dimensional complex vector space. It is nonzero in exactly one degree, namely the degree equal to the rank of the hyperplane arrangement.

math.AT↗
arXiv2009

Odd-dimensional Charney-Davis conjecture

More than once we have heard that the Charney-Davis Conjecture makes sense only for odd-dimensional spheres. This is to point out that in fact it is also a statement about even-dimensional spheres.

math.CO↗
arXiv2009

Cohomology of Coxeter groups with group ring coefficients: II

For any Coxeter group W, we define a filtration of H^*(W;ZW) by W-submodules and then compute the associated graded terms. More generally, if U is a CW complex on which W acts as a reflection group we compute the associated graded terms for H_*(U) and, in the case where the action is proper and cocompact, for H^*_c(U).

math.GR↗
arXiv2006

Totally real immersions of surfaces

Totally real immersions $f$ of a closed real surface $Σ$ in an almost complex surface $M$ are completely classified, up to homotopy through totally real immersions, by suitably defined homotopy classes $\frak{M}(f)$ of mappings from $Σ$ into a specific real 5-manifold $E(M)$, while $\frak{M}(f)$ themselves are subject to a single cohomology constraint. This follows from Gromov's observation that totally real immersions satisfy the h-principle. For the receiving complex surfaces $C^2$, $CP^1\times CP^1$, $CP^2$ and $CP^2 # m\bar{CP^2}$, $m=1,2,...,7$, and all $Σ$ (or, $CP^2 # 8\bar{CP^2}$ and all orientable $Σ$), we illustrate the above nonconstructive result with explicit examples of immersions realizing all possible equivalence classes. We also determine which equivalence classes contain totally real embeddings, and provide examples of such embeddings for all classes that contain them.

math.DG↗
arXiv2002

Immersions of surfaces in spin$^c$-manifolds with Higgs fields

We define a `Higgs field' for a four-dimensional spin$^c$-manifold to be a smooth section of its positive half-spinor bundle, transverse to the zero section, and defined only up to a positive functional factor. This is intended to be a generalization of almost complex structures on real four-manifolds, each of which may in fact be treated as a Higgs field without zeros for a specific spin$^c$-structure. The notions of totally real or pseudoholomorphic immersions of real surfaces in an almost complex manifold of real dimension four have straighforward generalizations to the case of a spin$^c$-manifold with a Higgs field. Our results consist, first, in showing that totally real immersions of closed oriented surfaces in four-dimensional spin$^c$-manifolds with Higgs fields have, basically, the same properties as in the almost-complex case, and, secondly, in providing a description of all pseudoholomorphic immersions of such surfaces in the four-sphere endowed with a "standard" Higgs field.

math.DG↗
arXiv2002

Characteristic classes of smooth fibrations

We construct characteristic classes of smooth (Hamiltonian) fibrations as as fiber integrals of products of Pontriagin (or Chern) classes of vertical vector bundles over the total space of the universal fibration. We give explicit formulae of these fiber integrals for toric manifolds and get estimates of the dimension of the cohomology groups of classifying spaces.

math.SG↗
arXiv2001

Commensurability of graph products

We define graph products of families of pairs of groups and study the question when two such graph products are commensurable. As an application we prove linearity of certain graph products.

math.GR↗
arXiv1999

New Kazhdan groups

The group of simplicial automorphisms of a Tits-Kac-Moody ininite building of thickness q associated to a cocompact reflexion group with fundamental domain a simplex, is Kazhdan for q sufficiently large. Thus we obtain families of new Kazhdan groups: two in dimension 3 and one in dimension 4. The proof uses continuos cohomology, in particular a lemma of Borel-Wallach, and Garland's vanishing method.

math.GR↗