Coincidence of maps on torus fiber bundles over the circle

João Peres Vieira

DOI: http://dx.doi.org/10.12775/TMNA.2015.057

Abstract


The main purpose of this work is to study coincidences of fibre-preserving self-maps over the circle $S^1$ for spaces which are fibre bundles over $S^1$ and the fibre is the torus $T$. We classify all pairs of self-maps over $S^1$ which can be deformed fibrewise to a pair of coincidence free maps.

Keywords


Coincidence; fibre bundle; fibrewise homotopy

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References


H.J. Baues, Obstruction Theory - on Homotopy Classification of Maps, Lecture Notes in Math., vol. 628, Springer, 1977.

R. Brooks, Coincidences, roots and fixed points, Doctoral Dissertation, University of California, Los Angeles, California, 1967.

A. Dold, The fixed point index of fiber-preserving maps, Invent. Math. 25 (1974), 281-297.

E. Fadell and S. Husseini, A fixed point theory for fiber-preserving maps, Lecture Notes in Mathematics, vol. 886, Springer, (1981), 49-72.

E. Fadell and S. Husseini, Fixed point theory for non simply connected manifolds, Topology, vol. 20, Springer, (1981), 53-92.

D. L. Goncalves, Fixed Point of $S^1$-Fibrations, Pacific J. Math. 129 (1987), 297-306.

D. L. Goncalves, Coincidence theory, Handbook of Topological Fixed Point Theory, Springer, (2005), 3-42.

D. L. Goncalves, D. Penteado and J. P. Vieira, Fixed Points on Torus Fiber Bundles over the Circle, Fund. Math. 183 (1) (2004), 1-38.

D. L. Goncalves, Fixed points on Klein bottle fiber bundles over the circle, Fund. Math. 203 (3) (2009), 263-292.

D. L. Goncalves, Abelianized Obstruction for fixed points of fiber-preserving maps of Surface bundles, Topol. Methods Nonlinear Anal. 33 (2) (2009), 293-305.

D. L. Johnson, Presentation of Groups, LMS Lecture Notes 22, Cambridge University Press (1976).

M. R. Kelly, Minimizing the number of fixed points for self-maps of compact surfaces, Pacific J. Math. 126 (1987), 81-123.

J. Nielsen, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flachen, Acta Math. 50 (1927), 189-358; English transl. in: Jakob Nielsen Collected Mathematical Papers, Birkhauser (1986), 223-341.

D. Penteado, Sobre Pontos Fixos de Aplicacoes entre Fibrados com Fibra Superficie, ICMSC-USP Sao Carlos- Sao Paulo (1988).

D. Penteado, Fixed points on surface fiber bundles, in: 10th Brazilian Topology Meeting, Mat. Contemp. 13 (1997), 251-267.

J. W. Vick, Homology Theory: An Introduction to Algebraic Topology, Academic Press (1973).

G. W. Whitehead, Elements of Homotopy Theory, Springer (1918).

P. Wong, Coincidence Theory for spaces which Fiber Over Nilmanifold, Hindawi Publishing Corporation Fixed Point theory and Applications (2004), 89-95.


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