### An effective criterion for the existence of a mass partition

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We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik–Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the ${\mathrm{\U0001d530\U0001d529}}_{2}$ Bernstein–Gelfand–Gelfand resolution as an Aomoto complex.

In this paper we compute topological invariants for some configuration spaces of complex projective spaces. We shall describe Sullivan models for these configuration spaces.

A bipolynomial is a holomorphic mapping of a sphere onto a sphere such that some point on the target sphere has exactly two preimages. The topological invariants of spaces of bipolynomials without multiple roots are connected with characteristic classes of rational functions with two poles and generalized braid groups associated to extended affine Weyl groups of the serie $A$. We prove that the cohomology rings of the spaces of bipolynomials of bidegree $(k,l)$ stabilize as $k$ tends to infinity and that...

We show that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2. We show that this estimate is optimal for the n-point configuration space of Γ if n ≥ 2b.

A closed form formula (generating function) for the Euler characteristic of the configuration space of n particles in a simplicial complex is given.

It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper we give an explicit formula of the Morse index for the signed area of a cyclic configuration. We show that it depends not only on the combinatorics of a cyclic configuration, but also on its metric properties.

Let ${\mathcal{F}}_{h}^{i}(k,n)$ be the $i$-th ordered configuration space of all distinct points ${H}_{1},...,{H}_{h}$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of ${\scriptstyle {\u2102}^{n}}$, whose sum is a subspace of dimension $i$. We prove that ${\mathcal{F}}_{h}^{i}(k,n)$ is (when non empty) a complex submanifold of $Gr{(k,n)}^{h}$ of dimension $i(n-i)+hk(i-k)$ and its fundamental group is trivial if $i=min(n,hk)$, $hk\ne n$ and $n\>2$ and equal to the braid group of the sphere ${\scriptstyle \u2102}$${P}^{}$

We prove that the space $ex{p}_{k}\bigvee {S}^{m+1}$ of nonempty subsets of cardinality at most k in a bouquet of m+1-dimensional spheres is (m+k-2)-connected. This, as shown by Tuffley, implies that the space $ex{p}_{k}X$ is (m+k-2)-connected for any m-connected cell complex X.

Following a Bendersky-Gitler idea, we construct an isomorphism between Anderson’s and Arone’s complexes modelling the chain complex of a mapping space. This allows us to apply Shipley’s convergence theorem to Arone’s model. As a corollary, we reduce the problem of homotopy equivalence for certain “toy” spaces to a problem in homological algebra.

We discuss some approaches to the topological study of real quadratic mappings. Two effective methods of computing the Euler characteristics of fibers are presented which enable one to obtain comprehensive results for quadratic mappings with two-dimensional fibers. As an illustration we obtain a complete topological classification of configuration spaces of planar pentagons.

We describe an alternative approach to some results of Vassiliev ([Va1]) on spaces of polynomials, by applying the "scanning method" used by Segal ([Se2]) in his investigation of spaces of rational functions. We explain how these two approaches are related by the Smale-Hirsch Principle or the h-Principle of Gromov. We obtain several generalizations, which may be of interest in their own right.