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Sep 21, 2013
09/13

by
Pavle V. M. Blagojevic

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\noindent The simultaneous partition problems are classical problems of the combinatorial geometry which have the natural flavor of the equivariant topology. The $k$-fan partition problems have attracted a lot of attention \cite{Aki2000}, \cite{BaMa2001}, \cite{BaMa2002} and forced some hard concrete combinatorial calculations in the equivariant cohomology \cite% {Bl-Vr-Ziv}. These problems can be reduced, by a beautiful scheme of \cite% {BaMa2001}, to a \textquotedblright...

Source: http://arxiv.org/abs/math/0402400v3

2
2.0

Jun 29, 2018
06/18

by
Pavle V. M. Blagojević; Roman Karasev

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Let $M$ and $N$ be smooth (real or complex) manifolds, and let $M$ be equipped with some Riemannian metric. A continuous map $f\colon M\longrightarrow N$ admits a local $k$-multiplicity if, for every real number $\omega >0$, there exist $k$ pairwise distinct points $x_1,\ldots,x_k$ in $M$ such that $f(x_1)=\cdots=f(x_k)$ and $\diam\{x_1,\ldots,x_k\}

Topics: Metric Geometry, Algebraic Topology, Mathematics

Source: http://arxiv.org/abs/1603.06723

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40

Sep 21, 2013
09/13

by
Pavle V. M. Blagojevic; Günter M. Ziegler

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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...

Source: http://arxiv.org/abs/0808.3841v1

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156

Sep 24, 2013
09/13

by
Pavle V. M. Blagojević; Günter M. Ziegler

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We describe a regular cell complex model for the configuration space F(\R^d,n). Based on this, we use Equivariant Obstruction Theory to 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.

Source: http://arxiv.org/abs/1202.5504v3

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30

Sep 18, 2013
09/13

by
Pavle V. M. Blagojević; Günter M. Ziegler

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We compute the complete Fadell-Husseini index of the 8 element dihedral group D_8 acting on S^d \times S^d, both for F_2 and for integer coefficients. This establishes the complete goup cohomology lower bounds for the two hyperplane case of Gr"unbaum's 1960 mass partition problem: For which d and j can any j arbitrary measures be cut into four equal parts each by two suitably-chosen hyperplanes in R^d? In both cases, we find that the ideal bounds are not stronger than previously...

Source: http://arxiv.org/abs/0704.1943v4

2
2.0

Jun 29, 2018
06/18

by
Pavle V. M. Blagojević; Roman Karasev; Alexander Magazinov

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A celebrated result of Dol'nikov, and of \v{Z}ivaljevi\'c and Vre\'cica, asserts that for every collection $\mu_1,\ldots,\mu_m$ of $m$ measures on the Euclidean space $\mathbb R^{n + m - 1}$ there exists a projection onto an $n$-dimensional vector subspace $\Gamma$ with a point in it at depth at least $\tfrac{1}{n + 1}$ with respect to each associated $n$-dimensional marginal measure $\Gamma_*\mu_1,\ldots,\Gamma_*\mu_m$. In this paper we consider a natural extension of this result and ask for a...

Topics: Metric Geometry, Combinatorics, Mathematics

Source: http://arxiv.org/abs/1606.08225

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126

Jul 20, 2013
07/13

by
Pavle V. M. Blagojevic; Aleksandra S. Dimitrijevic Blagojevic

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A significant group of problems coming from the realm of Combinatorial Geometry can only be approached through the use of Algebraic Topology. From the first such application to Kneser's problem in 1978 by Lov% \'{a}sz \cite{Lovasz} through the solution of the Lov\'{a}sz conjecture \cite% {Babson-Kozlov}, \cite{Carsten}, many methods from Algebraic Topology have been used. Specifically, it appears that the understanding of equivariant theories is of the most importance. The solution of many...

Source: http://arxiv.org/abs/math/0609053v2

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Sep 20, 2013
09/13

by
Pavle V. M. Blagojević; Wolfgang Lück; Günter M. Ziegler

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We study the Fadell-Husseini index of the configuration space F(R^d,n) with respect to different subgroups of the symmetric group S_n. For p prime and d>0, we completely determine Index_{Z/p}(F(R^d,p);F_p) and partially describe Index{(Z/p)^k}(F(R^d,p^k);F_p). In this process we obtain results of independent interest, including: (1) an extended equivariant Goresky-MacPherson formula, (2) a complete description of the top homology of the partition lattice Pi_p as an F_p[Z_p]-module, and (3) a...

Source: http://arxiv.org/abs/1207.2852v2

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Jul 20, 2013
07/13

by
Pavle V. M. Blagojević; Benjamin Matschke; Günter M. Ziegler

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We prove that any continuous map of an N-dimensional simplex Delta_N with colored vertices to a d-dimensional manifold M must map r points from disjoint rainbow faces of Delta_N to the same point in M: For this we have to assume that N \geq (r-1)(d+1), no r vertices of Delta_N get the same color, and our proof needs that r is a prime. A face of Delta_N is a rainbow face if all vertices have different colors. This result is an extension of our recent "new colored Tverberg theorem", the...

Source: http://arxiv.org/abs/1107.1904v1

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36

Sep 19, 2013
09/13

by
Pavle V. M. Blagojević; Benjamin Matschke; Günter M. Ziegler

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We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Barany et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Barany & 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.

Source: http://arxiv.org/abs/0910.4987v2

2
2.0

Jun 28, 2018
06/18

by
Pavle V. M. Blagojević; Florian Frick; Günter M. Ziegler

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Using the authors' 2014 "constraints method," we give a short proof for a 2015 result of Dobbins on representations of a point in a polytope as the barycenter of points in a skeleton, and show that the "r-fold Whitney trick" of Mabillard and Wagner (2014/2015) implies that the Topological Tverberg Conjecture for r-fold intersections fails dramatically for all r that are not prime powers.

Topics: Combinatorics, Algebraic Topology, Mathematics, Metric Geometry

Source: http://arxiv.org/abs/1510.07984

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3.0

Jun 30, 2018
06/18

by
Pavle V. M. Blagojević; Florian Frick; Günter M. Ziegler

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We give a short and simple proof of a recent result of Dobbins that any point in an $nd$-polytope is the barycenter of $n$ points in the $d$-skeleton. This new proof builds on the constraint method that we recently introduced to prove Tverberg-type results.

Topics: Mathematics, Metric Geometry, Combinatorics

Source: http://arxiv.org/abs/1411.4417

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35

Sep 22, 2013
09/13

by
Pavle V. M. Blagojevic; Sinisa T. Vrecica; Rade T. Zivaljevic

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We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of calculating the obstructions for the existence of equivariant maps. A variety of techniques are introduced and discussed with the emphasis on concrete and explicit calculations. This eventually leads (Theorems 18 and 19) to an almost exhaustive analysis of when such...

Source: http://arxiv.org/abs/math/0403161v1

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Sep 20, 2013
09/13

by
Pavle V. M. Blagojevic; Aleksandra S. Dimitrijevic Blagojevic; John McCleary

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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 $\mathbb{R}^m$ for which a coincidence of the image of this mapping with an arrangement $\mathcal{A}$ 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...

Source: http://arxiv.org/abs/math/0612002v2

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4.0

Jun 26, 2018
06/18

by
Pavle V. M. Blagojevic; Florian Frick; Albert Haase; Günter M. Ziegler

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In 1960 Gr\"unbaum asked whether for any finite mass in $\mathbb{R}^d$ there are $d$ hyperplanes that cut it into $2^d$ equal parts. This was proved by Hadwiger (1966) for $d\le3$, but disproved by Avis (1984) for $d\ge5$, while the case $d=4$ remained open. More generally, Ramos (1996) asked for the smallest dimension $\Delta(j,k)$ in which for any $j$ masses there are $k$ affine hyperplanes that simultaneously cut each of the masses into $2^k$ equal parts. At present the best lower...

Topics: Mathematics, Algebraic Topology

Source: http://arxiv.org/abs/1502.02975

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4.0

Jun 29, 2018
06/18

by
Pavle V. M. Blagojević; Aleksandra S. Dimitrijević Blagojević; Günter M. Ziegler

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We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer $D\geq 1$ and any collection of sets $\Gamma_1,\ldots,\Gamma_j$ of low-degree $k$-dimensional varieties in $\mathbb{R}^n$ there exists a non-zero polynomial $p\in\mathbb{R}[X_1,\ldots,X_n]$ of degree at most $D$ so that each connected component of $\mathbb{R}^n{\setminus}Z(p)$ intersects $O(jD^{k-n}|\Gamma_i|)$ varieties of...

Topics: Combinatorics, Algebraic Topology, Mathematics

Source: http://arxiv.org/abs/1601.01629

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10.0

Jun 28, 2018
06/18

by
Pavle V. M. Blagojević; Florian Frick; Albert Haase; Günter M. Ziegler

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The Gr\"unbaum-Hadwiger-Ramos hyperplane mass partition problem was introduced by Gr\"unbaum (1960) in a special case and in general form by Ramos (1996). It asks for the "admissible" triples $(d,j,k)$ such that for any $j$ masses in $\mathbb{R}^d$ there are $k$ hyperplanes that cut each of the masses into $2^k$ equal parts. Ramos' conjecture is that the Avis-Ramos necessary lower bound condition $dk\ge j(2^k-1)$ is also sufficient. We develop a "join scheme" for...

Topics: Metric Geometry, Combinatorics, Algebraic Topology, Mathematics

Source: http://arxiv.org/abs/1509.02959

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2.0

Jun 29, 2018
06/18

by
Pavle V. M. Blagojević; Aleksandra S. Dimitrijević Blagojević; Günter M. Ziegler

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In this paper we use the strength of the constraint method in combination with a generalized Borsuk-Ulam type theorem and a cohomological intersection lemma to show how one can obtain many new topological transversal theorems of Tverberg type. In particular, we derive a topological generalized transversal Van Kampen-Flores theorem and a topological transversal weak colored Tverberg theorem.

Topics: Metric Geometry, Combinatorics, Algebraic Topology, Mathematics

Source: http://arxiv.org/abs/1604.02814

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4.0

Jun 30, 2018
06/18

by
Pavle V. M. Blagojević; Frederick R. Cohen; Wolfgang Lück; Günter M. Ziegler

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A continuous map C^d -> C^N is a complex k-regular embedding if any k pairwise distinct points in C^d are mapped by f into k complex linearly independent vectors in C^N. Our central result on complex k-regular embeddings extends results of Cohen & Handel (1978), Chisholm (1979) and Blagojevic, L\"uck & Ziegler (2013) on real k-regular embeddings: We give new lower bounds for the existence of complex k-regular embeddings. These are obtained by modifying the framework of Cohen...

Topics: Mathematics, Algebraic Topology

Source: http://arxiv.org/abs/1410.6052