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Denise de Mattos

Publications and source records attributed to Denise de Mattos.

7 recordsLinked to original sources
arXiv2026

Equivariant category and Topological complexity of wedges

We prove the formula \begin{equation*} \text{cat}_G(X\vee Y)=\max\{\text{cat}_G(X),\text{cat}_G(Y)\} \end{equation*} for the equivariant category of the wedge $X\vee Y$. As a direct application, we have that the wedge $\bigvee_{i=1}^m X_i$ is $G$-contractible if and only if each $X_i$ is $G$-contractible, for each $i=1,\ldots,m$. One further application is to compute the equivariant category of the quotient $X/A$, for a $G$-space $X$ and an invariant subset $A$ such that the inclusion $A\hookrightarrow X$ is $G$-homotopic to a constant map $\overline{x_0}:A\to X$, for some $x_0\in X^G$. Additionally, we discuss the equivariant and invariant topological complexities for wedges. For instance, as applications of our results, we obtain the following equalities: \begin{align*} \text{TC}_G(X\vee Y)&=\max\{\text{TC}_G(X),\text{TC}_G(Y),\text{cat}_G(X\times Y)\}, \text{TC}^G(X\vee Y)&=\max\{\text{TC}^G(X),\text{TC}^G(Y),_{X\vee Y}\text{cat}_{G\times G}(X\times Y)\}, \end{align*} for $G$-connected $G$-CW-complexes $X$ and $Y$ under certain conditions. Keywords: (Equivariant) Lusternik-Schnirelmann category, equivariant and invariant topological complexities, $G$-spaces, wedge product, smash product

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arXiv2022

Colored Tverberg Theorems for non-prime powers

We prove a relative of the Optimal (Type B)} Colored Tverberg theorem of Živaljević and Vrećica which modifies this results in two different ways. (1) Our result is valid if the number of rainbow faces is $q= p^n-1$, where $p$ is a prime. (2) The size of rainbow simplices satisfies the condition $\vert C_i\vert \in \{2q-2, 2q+1\}$ while in the original theorem $\vert C_i\vert = 2q-1$ for all $i$.

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arXiv2022

Some results about Colored Tverberg Theorem

In this paper, we present some results related to Barany-Larman colored problem and The Zivaljevic and Vrecica colored Tverberg problem. We give an alternative proof for the Barany-Larman Conjecture for primes -1 and the optimal colored Tverberg theorem. Also, we prove generalizations of the Colored Tverberg theorem of Zivaljevic and Vrecica.

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arXiv2020

On the length of cohomology spheres

We present the length, a numerical cohomological index theory, of $ G $-spaces which are cohomology spheres and $ G $ is a $p$-torus or a torus group, where $p$ is a prime. As a consequence, we obtain Borsuk-Ulam and Bourgin-Yang type theorems in this context. A sharper version of the Bourgin-Yang theorem for topological manifolds is also proved. Also, we give some general results regarding the upper and lower bound for the length.

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arXiv2016

$(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$

Let $X$ be a paracompact space, let $G$ be a finite group acting freely on $X$ and let $H$ a cyclic subgroup of $G$ of prime order $p$. Let $f:X\rightarrow M$ be a continuous map where $M$ is a connected $m$-manifold (orientable if $p>2$) and $f^* (V_k) = 0$, for $k\geq 1$, where $V_k$ are the $Wu$ classes of $M$. Suppose that ${\rm{ind}}\, X\geq n> (|G|-r)m$, where $r=\frac{|G|}{p}$. In this work, we estimate the cohomological dimension of the set $A(f,H,G)$ of $(H,G)$-coincidence points of $f$. Also, we estimate the index of a $(H, G)$-coincidence set in the case that $H$ is a $p$-torus subgroup of a particular group $G$ and as application we prove a topological Tverberg type theorem for any natural number $r$. Such result is a weak version of the famous topological Tverberg conjecture, which was proved recently, fail for all $r$ that are not prime powers. Moreover, we obtain a generalized Van Kampen-Flores type theorem for any natural number $r$.

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arXiv2016

Bourgin-Yang versions of the Borsuk-Ulam theorem for $p$-toral groups

Let $V$ and $W$ be orthogonal representations of $G$ with $V^G= W^G=\{0\}$. Let $S(V )$ be the sphere of $V$ and $f : S(V ) \to W$ be a $G$-equivariant mapping. We give an estimate for the dimension of the set $Z_f=f^{-1}\{0\}$ in terms of $ \dim V$ and $\dim W$, if $G$ is the torus $\mathbb T^k$, or the $p$-torus $\mathbb Z_p^k$. This extends the classical Bourgin-Yang theorem onto this class of groups. Finally, we show that for any $p$-toral group $G$ and a $G$-map $f:S(V) \to W$, with $\dim V=\infty$ and $\dim W<\infty$, we have $\dim Z_f= \infty$.

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