Generalized Lefschetz numbers of pushout maps defined on non-connected spaces

Ferrario, Davide

Banach Center Publications, Tome 50 (1999), p. 117-135 / Harvested from The Polish Digital Mathematics Library

Résumé

Let A, $X_{1}$ and $X_{2}$ be topological spaces and let $i_{1} : A \to X_{1}$ , $i_{2} : A \to X_{2}$ be continuous maps. For all self-maps $f_{A} : A \to A$ , $f_{1} : X_{1} \to X_{1}$ and $f_{2} : X_{2} \to X_{2}$ such that $f_{1} i_{1} = i_{1} f_{A}$ and $f_{2} i_{2} = i_{2} f_{A}$ there exists a pushout map f defined on the pushout space $X_{1} ⊔_{A} X_{2}$ . In [F] we proved a formula relating the generalized Lefschetz numbers of f, $f_{A}$ , $f_{1}$ and $f_{2}$ . We had to assume all the spaces involved were connected because in the original definition of the generalized Lefschetz number given by Husseini in [H] the space was assumed to be connected. So, to extend the result of [F] to the not necessarily connected case, a definition of generalized Lefschetz number for a map defined on a not necessarily connected space is given; it reduces to the original one when the space is connected and it is still a trace-like quantity. It allows us to prove the pushout formula in this more general case and therefore to get a tool for computing Nielsen and generalized Lefschetz numbers in a wide class of spaces.

Publié le : 1999-01-01

Zbl 0939.55001

EUDML-ID : urn:eudml:doc:208954

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     author = {Ferrario, Davide},
     title = {Generalized Lefschetz numbers of pushout maps defined on non-connected spaces},
     journal = {Banach Center Publications},
     volume = {50},
     year = {1999},
     pages = {117-135},
     zbl = {0939.55001},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv49i1p117bwm}
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Ferrario, Davide. Generalized Lefschetz numbers of pushout maps defined on non-connected spaces. Banach Center Publications, Tome 50 (1999) pp. 117-135. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv49i1p117bwm/

Bibliographie

[000] [B] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott Foresman and Company, Chicago, 1971. | Zbl 0216.19601

[001] [FH] E. Fadell and S. Husseini, The Nielsen Number on Surfaces, Contemp. Math. 21, AMS, Providence, 1983. | Zbl 0563.55001

[002] [F] D. Ferrario, Generalized Lefschetz numbers of pushout maps, Topology Appl. 68 (1996) 67-81. | Zbl 0845.55003

[003] [H] S. Y. Husseini, Generalized Lefschetz Numbers, Trans. Amer. Math. Soc. 272 (1982), 247-274. | Zbl 0507.55001

[004] [J] B. J. Jiang, Lectures on Nielsen fixed point theory, Contemp. Math. 14, Amer. Math. Soc., Providence, 1983. | Zbl 0512.55003

[005] [J1] B. J. Jiang, Periodic orbits on surfaces via Nielsen fixed point theory, in: Topology Hawaii (Honolulu, HI, 1990), 101-118.

[006] [P] R. A. Piccinini, Lectures on Homotopy Theory, North-Holland, Amsterdam, 1992. | Zbl 0742.55001

[007] [S] J. Stallings, Centerless groups - an algebraic formulation of Gottlieb's theorem, Topology 4 (1965), 129-134. | Zbl 0201.36001