Pavle V. M. Blagojević

Equivariant topology of configuration spaces

We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors which induce weak equivalences on morphism objects and equivalences of ordinary categories when we take sets of connected components on morphism objects.

Turán numbers for K s,t -free graphs: Topological obstructions and algebraic constructions

We show that every hypersurface in ℝs × ℝs contains a large grid, i.e., the set of the form S × T, with S, T ⊂ ℝs. We use this to deduce that the known constructions of extremal K2,2-free and K3,3-free graphs cannot be generalized to a similar construction of Ks,s-free graphs for any s ≥ 4. We also give new constructions of extremal Ks,t-free graphs for large t.

Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story

Barany’s “topological Tverberg conjecture” from 1976 states that any continuous map of an N-simplex \(\Delta _{N}\) to \(\mathbb{R}^{d}\), for N ≥ (d + 1)(r − 1), maps points from r disjoint faces in \(\Delta _{N}\) to the same point in \(\mathbb{R}^{d}\). The proof of this result for the case when r is a prime, as well as some colored version of the same result, using the results of Borsuk–Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matousek’s 2003 book “Using the Borsuk–Ulam theorem.”

Computational Topology of Equivariant Maps from Spheres to Complements of Arrangements

The problem of the existence of an equivariant map is a classical topological problem ubiquitous in topology and its applications. Many problems in discrete geometry and combinatorics have been reduced to such a question and many of them resolved by the use of equivariant obstruction theory. A variety of concrete techniques for evaluating equivariant obstruction classes are introduced, discussed and illustrated by explicit calculations. The emphasis is on D 2n -equivariant maps from spheres to complements of arrangements, motivated by the problem of finding a 4-fan partition of 2-spherical measures, where D 2n is the dihedral group. One of the technical highlights is the determination of the D 2n -module structure of the homology of the complement of the appropriate subspace arrangement, based on the geometric interpretation for the generators of the homology groups of arrangements.

On highly regular embeddings

A continuous map \(\mathbb{R}^{d} \rightarrow \mathbb{R}^{N}\) is k-regular if it maps any k pairwise distinct points to k linearly independent vectors. Our main result on k-regular maps is the following lower bound for the existence of such maps between Euclidean spaces, in which α(k) denotes the number of ones in the dyadic expansion of k:

Symmetric products of surfaces and the cycle index

AbstractWe study some of the combinatorial structures related to the signature ofG-symmetric products of (open) surfacesSPGm(M)=Mm/G whereG ⊂Sm.The attention is focused on the question, what information about a surfaceM can be recovered from a symmetric productSPn(M). The problem is motivated in part by the study of locally Euclidean topological commutative (m+k,m)-groups, [16]. Emphasizing a combinatorial point of view we express the signature Sign(SPGm(M))in terms of the cycle index

Thieves can make sandwiches

Z\left( {G:\bar x} \right)

Index of Grassmann manifolds and orthogonal shadows

ofG, a polynomial which originally appeared in Pólya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfacesMg,k,Mg′,k′such that the manifoldsSPm(Mg,k)andSPm(M)g′,k′)are often not homeomorphic, although they always have the same homotopy type provided 2g+k=2g′+k′ andk,k′≥1.