Mostrando recursos 1 - 20 de 68

  1. Periodic points on T-fiber bundles over the circle

    Silva, Weslem Liberato; de Souza, Rafael Moreira

  2. Periodic points on T-fiber bundles over the circle

    Silva, Weslem Liberato; de Souza, Rafael Moreira

  3. The equivalence of two methods: finding representatives of non--empty Nielsen classes

    Hart, Evelyn L.; Vu, Ha T.
    Let $f:X\to X$ be a self--map with $X$ a wedge of circles or a compact surface with boundary, so that the fundamental group of $X$ is finitely generated and free. In [3], Wagner presents an algorithm for extracting information from the homomorphism induced by $f$ on the fundamental group. This information involves the fixed point index of $f$ and the Nielsen classes of fixed points of $f$. The step in which the representatives of Nielsen classes, Wagner tails, are calculated is equivalent to a step in the method presented by Fadell and Husseini in [1]. The Fadell--Husseini method was designed...

  4. The equivalence of two methods: finding representatives of non--empty Nielsen classes

    Hart, Evelyn L.; Vu, Ha T.
    Let $f:X\to X$ be a self--map with $X$ a wedge of circles or a compact surface with boundary, so that the fundamental group of $X$ is finitely generated and free. In [3], Wagner presents an algorithm for extracting information from the homomorphism induced by $f$ on the fundamental group. This information involves the fixed point index of $f$ and the Nielsen classes of fixed points of $f$. The step in which the representatives of Nielsen classes, Wagner tails, are calculated is equivalent to a step in the method presented by Fadell and Husseini in [1]. The Fadell--Husseini method was designed...

  5. Common value pairs and their estimations

    Gu, Ying; Zhao, Xuezhi
    We shall give a new treatment to intersection points of two maps, named common value pairs. Given two maps $f,g\colon X\to Y$. Instead of considering intersection points on target space $Y$, we focus on the pairs in the domains $X$, the pair $(u,v)$ with $f(u)=g(v)$. The set of all these pairs is exactly the preimage of product $f\times g$ at the diagonal in $Y^2$. We shall apply the idea of Nielsen root theory into such a general case: preimage of a set. Hence, some estimation for common value pairs and therefore for intersection points are obtained.

  6. Common value pairs and their estimations

    Gu, Ying; Zhao, Xuezhi
    We shall give a new treatment to intersection points of two maps, named common value pairs. Given two maps $f,g\colon X\to Y$. Instead of considering intersection points on target space $Y$, we focus on the pairs in the domains $X$, the pair $(u,v)$ with $f(u)=g(v)$. The set of all these pairs is exactly the preimage of product $f\times g$ at the diagonal in $Y^2$. We shall apply the idea of Nielsen root theory into such a general case: preimage of a set. Hence, some estimation for common value pairs and therefore for intersection points are obtained.

  7. Nielsen numbers of iterates and Nielsen type periodic numbers of periodic maps on tori and nilmanifolds

    Heath, Philip R.
    In this paper we compute the Nielsen numbers $N(f^m)$ and the Nielsen type numbers $NP_m(f)$ and $N\Phi_m(f)$ {\it for all $m$}, for periodic maps $f$ on tori and nilmanifolds. For fixed $m$, there are known formulas for these numbers for arbitrary maps on tori and nilmanifolds. However when seeking to determine these numbers for all $m$ for periodic maps, fascinating patterns and shortcuts are revealed. Our method has two main thrusts. Firstly we study $N(f^m)$, $NP_m(f)$ and $N\Phi_m(f)$ on primitives (maps whose linearizations consist of primitive roots of unity), and then secondly we employ fibre techniques to give an inductive...

  8. Nielsen numbers of iterates and Nielsen type periodic numbers of periodic maps on tori and nilmanifolds

    Heath, Philip R.
    In this paper we compute the Nielsen numbers $N(f^m)$ and the Nielsen type numbers $NP_m(f)$ and $N\Phi_m(f)$ {\it for all $m$}, for periodic maps $f$ on tori and nilmanifolds. For fixed $m$, there are known formulas for these numbers for arbitrary maps on tori and nilmanifolds. However when seeking to determine these numbers for all $m$ for periodic maps, fascinating patterns and shortcuts are revealed. Our method has two main thrusts. Firstly we study $N(f^m)$, $NP_m(f)$ and $N\Phi_m(f)$ on primitives (maps whose linearizations consist of primitive roots of unity), and then secondly we employ fibre techniques to give an inductive...

  9. Fixed point index bounds for self-maps on closed surfaces

    Gonçalves, D.L.; Kelly, M.R.
    Given a surface with non-positive Euler characteristic and non-empty boundary, and a map which has the least number of fixed points possible within its homotopy class there are known bounds (both upper and lower) regarding the fixed point indices of the map. This paper gives a new proof of this result. In addition, a relative version of the method is developed, which is then used to establish the same index bounds for the case of a closed surface of negative Euler characteristic.

  10. Fixed point index bounds for self-maps on closed surfaces

    Gonçalves, D.L.; Kelly, M.R.
    Given a surface with non-positive Euler characteristic and non-empty boundary, and a map which has the least number of fixed points possible within its homotopy class there are known bounds (both upper and lower) regarding the fixed point indices of the map. This paper gives a new proof of this result. In addition, a relative version of the method is developed, which is then used to establish the same index bounds for the case of a closed surface of negative Euler characteristic.

  11. A note on nontrivial intersection for selfmaps of complex Grassmann manifolds

    Monis, Thaís F. M.; Penteado, Northon C. L.; Ura, Sérgio T.; Wong, Peter
    Let $G(k,n)$ be the complex Grassmann manifold of $k$-planes in $\mathbb C^{k+n}$. In this note, we show that for $1< k

  12. A note on nontrivial intersection for selfmaps of complex Grassmann manifolds

    Monis, Thaís F. M.; Penteado, Northon C. L.; Ura, Sérgio T.; Wong, Peter
    Let $G(k,n)$ be the complex Grassmann manifold of $k$-planes in $\mathbb C^{k+n}$. In this note, we show that for $1< k

  13. Maps between Sol $3$-manifolds and coincidence Nielsen numbers

    Panzarin, Karen Regina
    Let $M_A$ be the torus bundle over $S^1$ obtained using as gluing map an Anosov matrix $A$. In this paper we discuss maps from $M_{A^r}$ to $M_A$ and compute the coincidence Nielsen numbers for such maps, moreover we use that such manifolds are double covers of torus semi-bundles and compute the coincidence Nielsen number for selfmaps of Sol $3$-manifolds which are torus semi-bundles.

  14. Maps between Sol $3$-manifolds and coincidence Nielsen numbers

    Panzarin, Karen Regina
    Let $M_A$ be the torus bundle over $S^1$ obtained using as gluing map an Anosov matrix $A$. In this paper we discuss maps from $M_{A^r}$ to $M_A$ and compute the coincidence Nielsen numbers for such maps, moreover we use that such manifolds are double covers of torus semi-bundles and compute the coincidence Nielsen number for selfmaps of Sol $3$-manifolds which are torus semi-bundles.

  15. Fixed point sets of equivariant fiber-preserving maps

    Souza, Rafael; Wong, Peter
    Given a selfmap $f:X\to X$ on a compact connected polyhedron $X$, H. Schirmer gave necessary and sufficient conditions for a nonempty closed subset $A$ to be the fixed point set of a map in the homotopy class of $f$. R. Brown and C. Soderlund extended Schirmer's result to the category of fiber bundles and fiber-preserving maps. The objective of this paper is to prove an equivariant analogue of Brown-Soderlund theorem result in the category of $G$-spaces and $G$-maps where $G$ is a finite group.

  16. Fixed point sets of equivariant fiber-preserving maps

    Souza, Rafael; Wong, Peter
    Given a selfmap $f:X\to X$ on a compact connected polyhedron $X$, H. Schirmer gave necessary and sufficient conditions for a nonempty closed subset $A$ to be the fixed point set of a map in the homotopy class of $f$. R. Brown and C. Soderlund extended Schirmer's result to the category of fiber bundles and fiber-preserving maps. The objective of this paper is to prove an equivariant analogue of Brown-Soderlund theorem result in the category of $G$-spaces and $G$-maps where $G$ is a finite group.

  17. Central configurations, Morse and fixed point indices

    Ferrario, D.L.
    We compute the fixed point index of non-degenerate central configurations for the $n$-body problem in the euclidean space of dimension $d$, relating it to the Morse index of the gravitational potential function $\bar U$ induced on the manifold of all maximal $O(d)$-orbits. In order to do so, we analyze the geometry of maximal orbit type manifolds, and compute Morse indices with respect to the mass-metric bilinear form on configuration spaces.

  18. Central configurations, Morse and fixed point indices

    Ferrario, D.L.
    We compute the fixed point index of non-degenerate central configurations for the $n$-body problem in the euclidean space of dimension $d$, relating it to the Morse index of the gravitational potential function $\bar U$ induced on the manifold of all maximal $O(d)$-orbits. In order to do so, we analyze the geometry of maximal orbit type manifolds, and compute Morse indices with respect to the mass-metric bilinear form on configuration spaces.

  19. Equivariant maps between representation spheres

    Błaszczyk, Zbigniew; Marzantowicz, Wacław; Singh, Mahender
    Let $G$ be a compact Lie group. We prove that if $V$ and $W$ are orthogonal $G$-representations such that $V^G=W^G=\{0\}$, then a $G$-equivariant map $S(V) \to S(W)$ exists provided that $\dim V^H \leq \dim W^H$ for any closed subgroup $H\subseteq G$. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when $G$ is a torus.

  20. Equivariant maps between representation spheres

    Błaszczyk, Zbigniew; Marzantowicz, Wacław; Singh, Mahender
    Let $G$ be a compact Lie group. We prove that if $V$ and $W$ are orthogonal $G$-representations such that $V^G=W^G=\{0\}$, then a $G$-equivariant map $S(V) \to S(W)$ exists provided that $\dim V^H \leq \dim W^H$ for any closed subgroup $H\subseteq G$. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when $G$ is a torus.

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