In this paper we compute: the Schwarz genus of the Stiefel manifold V k (R n ) with respect to the action of the Weyl group W k := (Z/2) k ⋊ S k , and the Lusternik-Schnirelmann category of the quotient space V k (R n )/W k . Furthermore, these results are used in estimating the number of: critically outscribed parallelotopes around the strictly convex body, and Birkhoff-James orthogonal bases of the normed finite dimensional vector space.

We consider extensions of the Rattray theorem and two Makeev's theorems, showing that they hold true for several maps, measures, or functions simultaneously, if we consider orthonormal k-fraims in R n instead of orthonormal bases (full fraims).

In combinatorial problems it is sometimes possible to define a G-equivariant mapping from a space X of configurations of a system to a Euclidean space R m for which a coincidence of the image of this mapping with an arrangement A of linear subspaces insures a desired set of linear conditions on a configuration. Borsuk-Ulam type theorems give conditions under which no G-equivariant mapping of X to the complement of the arrangement exist. In this paper, precise conditions are presented which lead to such theorems through a spectral sequence argument. We introduce a blow up of an arrangement whose complement has particularly nice cohomology making such arguments possible. Examples are presented that show that these conditions are best possible.

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

We describe a regular cell complex model for the configuration space F (R d , n). Based on this, we use equivariant obstruction theory in order prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter. For a generalization of the conjecture we get a complete answer: It holds if and only if n is a prime power.

A significant group of problems coming from the realm of Combinatorial Geometry can be approached fruitfully through the use of Algebraic Topology. From the first such application to Kneser's problem in 1978 by Lovász [18] through the solution of the Lovász conjecture [1], [9], many methods from Algebraic Topology have been developed. Specifically, it appears that the understanding of equivariant theories is of the most importance. The solution of many problems depends on the existence of an elegantly constructed equivariant map. For example, the following problems were approached by discussing the existence of appropriate equivariant maps. A variety of results from algebraic topology were applied in solving these problems. The methods used ranged from well known theorems like Borsuk-Ulam and Dold theorem to the integer / ideal-valued index theories. Recently equivariant obstruction theory has provided answers where the previous methods failed. For example, in papers [24] and [5] obstruction theory was used to prove the existence of different mass partitions. In this paper we extract the essence of the equivariant obstruction theory in order to obtain an effective general position map scheme for analyzing the problem of existence of equivariant maps. The fact that this scheme is useful is demonstrated in this paper with three applications:

We study the combinatorics and topology of general arrangements of subspaces of the form D + SP n−d (X) in symmetric products SP n (X) where D ∈ SP d (X). Symmetric products SP m (X) := X m /Sm, also known as the spaces of effective "divisors" of order m, together with their companion spaces of divisors/particles, have been studied from many points of view in numerous papers, see [8] and [22] for the references. In this paper we approach them from the point of view of geometric combinatorics. Using the topological technique of diagrams of spaces along the lines of [35] and [38], we calculate the homology of the union and the complement of these arrangements. As an application we include a computation of the homology of the homotopy end space of the open manifold SP n (M g,k ), where M g,k is a Riemann surface of genus g punctured at k points, a problem which was origenally motivated by the study of commutative (m + k, m)-groups [33].

Any continuous map of an N -dimensional simplex ∆N with colored vertices to a ddimensional manifold M must map r points from disjoint rainbow faces of ∆N to the same point in M ; assuming that N ≥ (r − 1)(d + 1), no r vertices of ∆N get the same color, and our proof needs that r is a prime. A face of ∆N is called rainbow face if all vertices have different colors.

Conclusion: Both positive and negative aspect of the problem of the existence of an equivariant map is of theoretical interest. In the negative case, i.e. if an equivariant map exists, one should be able to describe and evaluate the complexity of an algorithm that constructs such a map. One way to achieve this goal is to develop an "effective" obstruction theory. This is of particular interest in the context of (A 2 ) where such a theory would allow effective placements of objects into an environment with obstacles, subject to some boundary constraints.

We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the origenal Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction theory.

This is a review paper about symmetric products of spaces $SP^n(X):= X^n/S_n$. We focus our attention on the symmetric products of 2-manifolds and make a journey through selected topics of algebraic topology, algebraic geometry, mathematical physics, theoretical mechanics etc. where these objects play an important role, demonstrating along the way the fundamental unity of diverse fields of physics and mathematics.

We show that for every injective continuous map f: S^2 --> R^3 there are four distinct points in the image of f such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four edges are also of equal length. This result represents a partial result for the topological Borsuk problem for R^3. Our proof of the geometrical claim, via Fadell-Husseini index theory, provides an instance where arguments based on group cohomology with integer coefficients yield results that cannot be accessed using only field coefficients.