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

Davide Ferrario

Banach Center Publications (1999)

  • Volume: 49, Issue: 1, page 117-135
  • ISSN: 0137-6934

Abstract

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Let A, X 1 and X 2 be topological spaces and let i 1 : A → X 1 , i 2 : A → X 2 be continuous maps. For all self-maps f A : A → A , f 1 : X 1 → X 1 and f 2 : X 2 → 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.

How to cite

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Ferrario, Davide. "Generalized Lefschetz numbers of pushout maps defined on non-connected spaces." Banach Center Publications 49.1 (1999): 117-135. <http://eudml.org/doc/208954>.

@article{Ferrario1999,
abstract = {Let A, $X_1$ and $X_2$ be topological spaces and let $i_1 : A → X_1$, $i_2: A → X_2$ be continuous maps. For all self-maps $f_A: A → A$, $f_1: X_1 → X_1$ and $f_2: X_2 → X_2$ such that $f_1i_1 = i_1f_A$ and $f_2i_2=i_2f_A$ 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.},
author = {Ferrario, Davide},
journal = {Banach Center Publications},
keywords = {Lefschetz number; Reidemeister number; pushout map},
language = {eng},
number = {1},
pages = {117-135},
title = {Generalized Lefschetz numbers of pushout maps defined on non-connected spaces},
url = {http://eudml.org/doc/208954},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Ferrario, Davide
TI - Generalized Lefschetz numbers of pushout maps defined on non-connected spaces
JO - Banach Center Publications
PY - 1999
VL - 49
IS - 1
SP - 117
EP - 135
AB - Let A, $X_1$ and $X_2$ be topological spaces and let $i_1 : A → X_1$, $i_2: A → X_2$ be continuous maps. For all self-maps $f_A: A → A$, $f_1: X_1 → X_1$ and $f_2: X_2 → X_2$ such that $f_1i_1 = i_1f_A$ and $f_2i_2=i_2f_A$ 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.
LA - eng
KW - Lefschetz number; Reidemeister number; pushout map
UR - http://eudml.org/doc/208954
ER -

References

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  1. [B] R. F. Brown, The Lefschetz Fixed Point Theorem, Scott Foresman and Company, Chicago, 1971. Zbl0216.19601
  2. [FH] E. Fadell and S. Husseini, The Nielsen Number on Surfaces, Contemp. Math. 21, AMS, Providence, 1983. Zbl0563.55001
  3. [F] D. Ferrario, Generalized Lefschetz numbers of pushout maps, Topology Appl. 68 (1996) 67-81. Zbl0845.55003
  4. [H] S. Y. Husseini, Generalized Lefschetz Numbers, Trans. Amer. Math. Soc. 272 (1982), 247-274. Zbl0507.55001
  5. [J] B. J. Jiang, Lectures on Nielsen fixed point theory, Contemp. Math. 14, Amer. Math. Soc., Providence, 1983. Zbl0512.55003
  6. [J1] B. J. Jiang, Periodic orbits on surfaces via Nielsen fixed point theory, in: Topology Hawaii (Honolulu, HI, 1990), 101-118. 
  7. [P] R. A. Piccinini, Lectures on Homotopy Theory, North-Holland, Amsterdam, 1992. Zbl0742.55001
  8. [S] J. Stallings, Centerless groups - an algebraic formulation of Gottlieb's theorem, Topology 4 (1965), 129-134. Zbl0201.36001

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