Livre-docência (09/2019):

Área: Topologia/Geometria - Topologia Algébrica.

Institute of Mathematics Casimir the Great University, Bydgoszcz, Polônia.

Área: Topologia/Geometria - Topologia Algébrica.

Supervisor: Prof. dr hab. Marek Golasinski.

Bolsa: FAPESP. Processo: 2014/21926-1

Wydzial Matematyki i Informatyki, Uniwersytetu Mikolaja Kopernika w Toruniu (Nicolaus Copernicus University), Torun, Polônia.

Área: Topologia/Geometria - Topologia Algébrica.

Supervisor: Prof. dr hab. Marek Golasinski.

Doutor em Matemática pela Universidade de São Paulo, USP, São Carlos.

Área: Topologia/Geometria - Topologia Algébrica.

Tese: 'Torção de Reidemeister das formas espaciais esféricas' (arquivo pdf).

Orientador: Prof. Dr. Mauro Flavio Spreafico.

Mestre em Matemática pela Universidade de São Paulo, USP, São Carlos.

Área: Topologia/Geometria - Topologia Algébrica.

Dissertação: 'Enumeração dos fibrados vetoriais sobre superfícies fechadas'. (arquivos pdf - dvi)

Orientador: Prof. Dr. Mauro Flavio Spreafico.

Bolsista PET (2000), com o projeto 'Visualização e redução das quádricas', em conjunto com o bolsista PET Daniel Ricardo Godoy, ambos orientados pela Profa. Dra. Alice Kimie Miwa Libardi.

Bolsista FAPESP de Iniciação Científica (2001-2002), com o projeto 'Coincidências antipodais para aplicações da esfera em complexos', orientado pela Profa. Dra. Alice Kimie Miwa Libardi.

### PesquisaResearch

Formas Espaciais Esféricas
Spherical Space Forms

inserir

Torção de Reidemeister
Reidemeister Torsion

inserir

Curvas em Espaços Homogêneos
Curves in Homogeneous Spaces

We consider $\Phi:G\times G/H \to G/H$ the natural action of the Lie group $G$ on the quotient space $G/H$, where $H\subset G$ is a closed subgroup. For $g\in G$ and $x\in G/H$ we sometimes write $\Phi(g,x)=g\cdot x$. Let $G_x$ be the isotropy subgroup at $x\in G/H$ and $\mathfrak{g}_x$ its Lie algebra. Then every $\mathbf{v}\in \mathfrak g$, $\mathbf{v}\notin \mathfrak g_x$, determines in $G/H$ the exponential curve $s\mapsto \exp(s\mathbf{v})\cdot x$ which is an embedding in some neighborhood of $s=0$. Now given an embedding $\alpha:I\to G/H$ of a real open interval $I$ into $G/H$, for each $t\in I$ one may consider the contact between $\alpha$ and $\beta_{\mathbf{v}}$ at $\alpha(t)=\beta_{\mathbf{v}}(0)$, $\beta_{\mathbf{v}}$ being the exponential curve $\beta_{\mathbf{v}}(s)=\exp(s\mathbf{v})\cdot \alpha(t)$. For the purpose of Geometry the nice notion of contact is as follows. We say that $\alpha$ and $\beta_\mathbf{v}$ are in contact of order $k$ at $\alpha(t)=\beta_{\mathbf{v}}(0)$ if some reparameterization of them have the same $k$-jet there.

Path-components of Map Spaces

Let $M(\mathbb{S}^m,\mathbb{S}^n)$ be the space of maps from the $m$-sphere $\mathbb{S}^m$ into the $n$-sphere $\mathbb{S}^n$ with $m,n\ge 1$. We estimate the number of homotopy types of path-components $M_\alpha(\mathbb{S}^{n+k},\mathbb{S}^n)$ and fibre homotopy types of evaluation fibrations $\omega_\alpha : M_\alpha(\mathbb{S}^{n+k},\S^n)\to \mathbb{S}^n$ for $8\le k\le 13$ and $\alpha\in\pi_{n+k}(\mathbb{S}^n)$, extending the results of [Golasiński (2009)]. Further, the number of strong homotopy types of $\omega_\alpha:M_\alpha(\mathbb{S}^{n+k},\mathbb{S}^n)\to\mathbb{S}^n$ for $8\le k\le 13$ is determined and some improvements of results from [Golasiński (2009)] are obtained.

Cyclic and cocyclic maps and generalized Whitehead products

The Gottlieb group $G_n(X)$ of a space $X$ is the subgroup of the homotopy group $\pi_n(X)$ of $X$ consisting of homotopy classes of maps $f : \mathbb{S}^n\to X$ such that the map $f\vee \operatorname{id}_X: \mathbb{S}^n\vee X\to X$ admits an extension $F: \mathbb{S}^n\times X\to X$. The study of the properties and structure of the Gottlieb groups represents a fundamental problem in homotopy theory dating back to their introduction by D. Gottlieb in the 1960's. Connections between the Gottlieb groups and fixed point theory, transformation groups, covering spaces and the homotopy theory of fibrations have been extensively researched. The definition of $G_n(X)$ uses the concept of cyclic homotopies. K. Varadarajan studies the role of cyclic and cocyclic (dual of cyclic) maps in the set-up of Eckmann-Hilton duality. The set of homotopy classes of cyclic maps $X\to Y$, denoted by $\mathcal{G}(X,Y)$ is a group provided $X$ carries an H-cogroup structure. Dually, the set of homotopy classes of cocyclic maps $X\to Y$, denoted by $\mathcal{DG}(X,Y)$ is a group provided $Y$ carries an H-group structure. Relationships between these generalized Gottlieb (dual Gottlieb groups) and the generalized Whitehead product (the dual generalized Whitehead product) have been considered. The aim of this paper is to present those results in the context of the so called Theriault product considered by B. Gray being an extended version of the generalized Whitehead product and its dual.

Produtos de Whitehead de ordem superior

We extend the notion of the exterior Whitehead product for maps $\alpha_i:\Sigma A_i \to X_i$ for $i=1,\dots,n$, where $\Sigma A_i$ is the reduced suspension of $A_i$ and then, for the interior product with $X_i=J_{m_i}(X)$ as well. The main result generalizes Theorem 1.10 from Hardie, K.A.: A generalization of the Hopf construction, Quart. J. Math. Oxford Ser. (2) 12 (1961), 196-204 and concerns to the Hopf invariant of the generalized Hopf construction. We close the paper applying Gray's construction $\circ$ (called the Theriault product) to a sequence $X_1,\dots,X_n$ of simply connected co-$H$-spaces to obtain a higher Gray-Whitehead product map $w_n:\Sigma^{n-2}(X_1\circ\dots\circ X_n)\to T_1(X_1,\dots,X_n),$ where $T_1(X_1,\dots,X_n)$ is the fat wedge of $X_1,\dots,X_n$.

### Publicações [ mybib.sh ]Publications

1. GOLASIŃSKI, M.; DE MELO, T.; SANTOS, E. L. Homotopies of maps of projective spaces and their cohomotopy groups. To appear in Topology and its Applications (2020)
2. BIASI, C.; LIBARDI, A.K.M.; DE MELO, T.; SANTOS, E.L. Some results on extension of maps and applications. Proc. Roy. Soc. Edinburgh Sect. A, v. 149 (6), 1465-1472, DOI 10.1017/prm.2018.113 (2019)
3. GOLASIŃSKI, M.; DE MELO, T. Generalized Gottlieb and Whitehead center groups of space forms. Homology, Homotopy and Applications, v. 21 (1), 323-340, DOI 10.4310/HHA.2019.v21.n1.a15 (2019).
4. GOLASIŃSKI, M.; DE MELO, T. Generalized Jiang and Gottlieb groups. Georgian Math. J., v. 25 (4), 523-528, DOI 10.1515/gmj-2018-0060 (2018).
5. GOLASIŃSKI, M.; DE MELO, T.; SANTOS, E. L. On path-components of the mapping spaces $M(\mathbb{S}^m,\mathbb{F}P^n)$. Manuscripta Mathematica, v. 158 (3-4), 401-419 (2019). DOI 10.1007/s00229-018-1012-5 (online 02/2018).
6. GOLASIŃSKI, M.; DE MELO, T. On the higher Whitehead product. Journal of Homotopy and Related Structures, v. 11 (4), 825-845, DOI 10.1007/s40062-016-0153-z (2016).
7. GOLASIŃSKI, M.; DE MELO, T. On the higher order exterior and interior Whitehead products. Manuscripta Mathematica, v. 152 (1), 167-188 (2017). DOI 10.1007/s00229-016-0857-8 (2016).
8. GOLASIŃSKI, M.; DE MELO, T. Evaluation fibrations and path-components of the mapping space $M(\mathbb{S}^{n+k},\mathbb{S}^n)$ for $8\le k\le 13$. Ukrainian Mathematical Journal, v. 65 (8), 1141-1154, DOI 10.1007/s11253-014-0849-3 (2014). [Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 8, pp. 1023–1034, August, 2013].
9. GOLASIŃSKI, M.; DE MELO, T. Cyclic and cocyclic maps and Generalized Whitehead products. Збірник праць Інститут математики Національної академії наук України, T. 6 (6), 22-34 (2013). [Zbìrnik prac' Ìnstitutu Matematiki NAN Ukraini]
10. NETO, O.M.; DE MELO, T.; SPREAFICO, M. Cellular decomposition of quaternionic spherical space forms. Geometriae Dedicata, v. 162 (1), 9-24, DOI 10.1007/s10711-012-9714-4 (2013).
11. DE MELO, T.; HARTMANN, L.; SPREAFICO, M. The analytic torsion of a disc. Annals of Global Analysis and Geometry, v. 42 (1), 29-59, DOI 10.1007/s10455-011-9300-2 (2012).
12. LIBARDI, A.K.M.;DE MELO, T.; VIEIRA, J.P. Invariantes Topológicos. 1a ed. São Paulo: Cultura Acadêmica, v1. 76p. (2012).
13. DE MELO, T.; HARTMANN, L.; SPREAFICO, M. Reidemeister torsion and analytic torsion of discs. Bollettino della Unione Matematica Italiana (1922) (Cessou em 1975. Desdobrado em dois: ISSN 0392-4033 Bollettino della Unione Matematica Italiana. A, v. 9, p. 529-533 (2009).
14. DE MELO, T.; SPREAFICO, M. Reidemeister torsion and analytic torsion of spheres. Journal of Homotopy and Related Structures, v. 04, p. 181-185 (2009).
15. DE MELO, T.; GRULHA JR, N. Sobre Variedades Polares de Superfícies. Matemática Universitária, v. 41, p. 13-26 (2006).
• PEREIRA, J.V.; DE MELO, T.; MIO, W.; VON ZUBEN, C.J.; LIBARDI, A.K.M. Optimization of Maggot Mass Rearing via Geometric Statistical Analysis (2018) [preprint].
• GOLASIŃSKI, M.; DE MELO, T. On homotopy and Gottlieb groups of some Moore spaces $M(A,n)$ (2017) [preprint].
• DE MELO, T.; DO NASCIMENTO, V. M.; SAEKI, O. Contact as applied to the geometry of curves in homogeneous spaces (2014) [preprint].