Coincidence Wecken Property for Nilmanifolds
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摘要
Let f, g: X → Y be maps from a compact infra-nilmanifold X to a compact nilmanifold Y with dim X ≥ dim Y. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number N(f, g) vanishes then f and g are deformable to be coincidence free. We also show that if X is a connected finite complex X and the Reidemeister coincidence number R(f, g) = ∞ then f ～ f' so that C(f', g) = {x ∈ X | f'(x) = g(x)} is empty.
Let f, g: X → Y be maps from a compact infra-nilmanifold X to a compact nilmanifold Y with dim X ≥ dim Y. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number N(f, g) vanishes then f and g are deformable to be coincidence free. We also show that if X is a connected finite complex X and the Reidemeister coincidence number R(f, g) = ∞ then f ～ f' so that C(f', g) = {x ∈ X | f'(x) = g(x)} is empty.
Let f, g: X → Y be maps from a compact infra-nilmanifold X to a compact nilmanifold Y with dim X ≥ dim Y. In this note, we show that a certain Wecken type property holds, i.e., if the Nielsen number N(f, g) vanishes then f and g are deformable to be coincidence free. We also show that if X is a connected finite complex X and the Reidemeister coincidence number R(f, g) = ∞ then f ～ f' so that C(f', g) = {x ∈ X | f'(x) = g(x)} is empty.
引文
[1]Brooks,R.B.S.:On removing coincidences of two maps when only one,rather than both,of them may be deformed by a homotopy.Pacific J.Math.,40,45-52(1972)
[2]Brooks,R.B.S.:On the sharpness of theΔ2 andΔ1 Nielsen numbers.J.Reine Angew.Math.,259,101-108(1973)
[3]Gon?calves,D.,Jezierski,J.,Wong,P.:Obstruction theory and coincidences in positive codimension.Acta Math.Sin.Engl.Ser.,22(5),1591-1602(2006)
[4]Gon?calves,D.,Wong,P.:Wecken property for roots.Proc.Amer.Math.Soc.,133(9),2779-2782(2005)
[5]Gon?calves,D.,Wong,P.:Obstruction theory and coincidences of maps between nilmanifolds.Archiv Math.,84,568-576(2005)
[6]Gon?calves,D.,Wong,P.:Nilmanifolds are Jiang-type spaces for coincidences.Forum Math.,13,133-141(2001)
[7]Gon?calves,D.,Wong,P.:Homogeneous spaces in coincidence theory II.Forum Math.,17,297-313(2005)
[8]Vendr′uscolo,D.,Wong,P.:Jiang-type theorems for coincidences of maps into homogeneous spaces.Topol.Methods Nonlinear Anal.,31,151-160(2008)
[2]Brooks,R.B.S.:On the sharpness of theΔ2 andΔ1 Nielsen numbers.J.Reine Angew.Math.,259,101-108(1973)
[3]Gon?calves,D.,Jezierski,J.,Wong,P.:Obstruction theory and coincidences in positive codimension.Acta Math.Sin.Engl.Ser.,22(5),1591-1602(2006)
[4]Gon?calves,D.,Wong,P.:Wecken property for roots.Proc.Amer.Math.Soc.,133(9),2779-2782(2005)
[5]Gon?calves,D.,Wong,P.:Obstruction theory and coincidences of maps between nilmanifolds.Archiv Math.,84,568-576(2005)
[6]Gon?calves,D.,Wong,P.:Nilmanifolds are Jiang-type spaces for coincidences.Forum Math.,13,133-141(2001)
[7]Gon?calves,D.,Wong,P.:Homogeneous spaces in coincidence theory II.Forum Math.,17,297-313(2005)
[8]Vendr′uscolo,D.,Wong,P.:Jiang-type theorems for coincidences of maps into homogeneous spaces.Topol.Methods Nonlinear Anal.,31,151-160(2008)