拓扑和的自同伦等价群
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摘要
如果拓扑空间X，Y的拓扑和X∨Y的自同伦等价可以对角化，则X∨Y的自同伦等价群Aut(X∨Y)可表示为它的两个子群Aut_x(X∨Y)与Aut_Y(X∨Y)的乘积。而且Aut(X∨Y)的特殊子群Aut_*(X∨Y)也有类似的结论。
The group of self-homotopy equivalences Aut(X V Y) is represented as a product of two subgroupsAutx(X\/Y)and Auty(XVY) under the assumption that the self-equivalences of X V Y can be diagonalized. Moreover , an analogous result holds for the special subgroup Aut*(X V Y) of Aut(X V Y).
The group of self-homotopy equivalences Aut(X V Y) is represented as a product of two subgroupsAutx(X\/Y)and Auty(XVY) under the assumption that the self-equivalences of X V Y can be diagonalized. Moreover , an analogous result holds for the special subgroup Aut*(X V Y) of Aut(X V Y).
引文
[1] Y.Ando and K.Yamaguchi,On the homotopy self-equivalences of the product A×B,Proc. Japan Acad,58(1982),323-325.
[2] M.Arkowitz,The group of self-homotopy equivalences-a survey, In Group of self-homotopy equivalences ang related topics (ed.R.A.Piccinini),Lecture Notes in Math ematics 1425 (Springer,1990),170-203.
[3] M.Arkowitz and C.R.Curjel,Groups of homotopy classes.Lecture Notes in Math 4, (Springer, 1964).
[4] M.Arkowitz and C.R.Curjel,The group of homotopy equivalences of a space, Bull. Amer. Math. Soc.70(1964),293-296.
[5] W.D.Barcus and M.G.Barratt,On the homotopy classification of the extensions of a fixed map,Trans. Amer. Math. Soc.88(1958),57-74.
[6] P. Booth and P. Heath,On the groups ξ(X×Y) and ξ_B~B(X×_BY), In Group of self-homotopy equivalences ang related topics (ed.R.A.Piccinini),Lecture Notes in Math ematics 1425 (Springer,1990),17-31.
[7] A.Dold,Partition of unity in the theory of fibrations,Ann. Math.78(1963), 223-255.
[8] D.Frank and D.W.Kahn,Finite complexes with infinitely-generated groups of self-equivalences,Topology 16(1977), 189-192.
[9] P. Heath. On the group ξ(X×Y) of Self-homotopy equivalences of a product, Quaest.Math.19(1996),433-451.
[10] P.J.Hilton and U.Stammbach,A course in homological algebra,(Springer, 1970).
[11] P.J.Hilton,Homotopy theory and duality, (New York: Cordon and Breach, 1965).
[12] I.M.James,General topology and homotopy theory, (Springer, 1984).
[13] D.W.Kahn, The groups of homotopy equivalences,Math. Z.84(1964), 1-8.
[14] D.W.Kahn,Some research problems on homotopy-self-equivalences, Lecture Notes in Math 1425.(Springer,1990),204-207.
[15] P.J.Kahn,Self-equivalences of (n-1)-connected 2n-manifolds,Math. Ann.180(1966),26-47.
[16] Y.Kudo and K.Tsuchida,On the generalized Barcus-Barratt sequence, Sci. Rep.Hirosaki Univ,13(1967),1-9.
[17] K.Maryuma and M.Mimura,On the group of self-homotopy equivalences of KP~2∨S~m,Mem. Fac. Sci. Kyushu Univ,38(1984),65-74.
[18] W.Metzler and A Zimmermann,selbstaquivalenzen von S~3×S~3 in quaternionisher Behandlung,Arch.der Math. ⅩⅦ(1971)，209-213.
[19] S.Oka,N.Sawashita and M.Sugawara,On the group of self-equivalences of a mappings cone,Hiroshima Math. J.4(1974),9-28.
[20] P. Pavesic,Self-homotopy equivalences of product spaces, Proceedings of the Royal Society of Edinburgh,129A(1999),181-197.
[21] J.Roitberg. Residually finite,Hopfian and coHopfian spaces,In Contemporary Mathematics 37(Providence,RI:AMS, 1985),131-144.
[22] J.Rutter,Homotopy classification of maps Between pseudo- projective spaces,Questiones Mathematicae,11(1988),409-422.
[23] J.W.Rutter,The groups of homotopy self equivalences classes of CW- complexes, Math. Proc. Camb. Phil. Soc,93 (1983),275-293.
[24] J.W.Rutter,Groups of self-homotopy equivalences of induced space,Comm. Math. Helv,45(1970),236-255.
[25] J.W.Rutter,Homotopy self-equivalences 1988-1999.In Groups of homoopy self-equivalences and related topics,(ed. Ken-ichi Maruyama and John W.Rutter),Contemporary Mathematics 274(2001),1-11.
[26] N.Sawashita, On the group of self-equivalences of the product of spheres. Hiroshima Math. J.5(1975),69-86.
[27] W.Shin,On the group ξ[X] of homotopy equivalence maps. Bull. Amer. Math. Soc.72 (1964),361-365.
[28] A.J.Sieradske,Twisted self-homotopy equivalences,Pacific J.Math.3 (1970),789-802.
[29] E.H.Spanier,Algebraic Topology,(Springer, 1966).
[30] G.W.Whitehead,Elements of homotopy theory, GTM 61 (Springer,1978).
[31] T.Yamanoshita,On the spaces of self-homotopy equivalences of certain CW-complexes,J.Math. Soc. Jpn 37(1985),455-470.
[32] K.Yamaguchi,On the self-homotopy of wedge of certain complexed,Kodai Math. J,6(1983), 1-30.
[2] M.Arkowitz,The group of self-homotopy equivalences-a survey, In Group of self-homotopy equivalences ang related topics (ed.R.A.Piccinini),Lecture Notes in Math ematics 1425 (Springer,1990),170-203.
[3] M.Arkowitz and C.R.Curjel,Groups of homotopy classes.Lecture Notes in Math 4, (Springer, 1964).
[4] M.Arkowitz and C.R.Curjel,The group of homotopy equivalences of a space, Bull. Amer. Math. Soc.70(1964),293-296.
[5] W.D.Barcus and M.G.Barratt,On the homotopy classification of the extensions of a fixed map,Trans. Amer. Math. Soc.88(1958),57-74.
[6] P. Booth and P. Heath,On the groups ξ(X×Y) and ξ_B~B(X×_BY), In Group of self-homotopy equivalences ang related topics (ed.R.A.Piccinini),Lecture Notes in Math ematics 1425 (Springer,1990),17-31.
[7] A.Dold,Partition of unity in the theory of fibrations,Ann. Math.78(1963), 223-255.
[8] D.Frank and D.W.Kahn,Finite complexes with infinitely-generated groups of self-equivalences,Topology 16(1977), 189-192.
[9] P. Heath. On the group ξ(X×Y) of Self-homotopy equivalences of a product, Quaest.Math.19(1996),433-451.
[10] P.J.Hilton and U.Stammbach,A course in homological algebra,(Springer, 1970).
[11] P.J.Hilton,Homotopy theory and duality, (New York: Cordon and Breach, 1965).
[12] I.M.James,General topology and homotopy theory, (Springer, 1984).
[13] D.W.Kahn, The groups of homotopy equivalences,Math. Z.84(1964), 1-8.
[14] D.W.Kahn,Some research problems on homotopy-self-equivalences, Lecture Notes in Math 1425.(Springer,1990),204-207.
[15] P.J.Kahn,Self-equivalences of (n-1)-connected 2n-manifolds,Math. Ann.180(1966),26-47.
[16] Y.Kudo and K.Tsuchida,On the generalized Barcus-Barratt sequence, Sci. Rep.Hirosaki Univ,13(1967),1-9.
[17] K.Maryuma and M.Mimura,On the group of self-homotopy equivalences of KP~2∨S~m,Mem. Fac. Sci. Kyushu Univ,38(1984),65-74.
[18] W.Metzler and A Zimmermann,selbstaquivalenzen von S~3×S~3 in quaternionisher Behandlung,Arch.der Math. ⅩⅦ(1971)，209-213.
[19] S.Oka,N.Sawashita and M.Sugawara,On the group of self-equivalences of a mappings cone,Hiroshima Math. J.4(1974),9-28.
[20] P. Pavesic,Self-homotopy equivalences of product spaces, Proceedings of the Royal Society of Edinburgh,129A(1999),181-197.
[21] J.Roitberg. Residually finite,Hopfian and coHopfian spaces,In Contemporary Mathematics 37(Providence,RI:AMS, 1985),131-144.
[22] J.Rutter,Homotopy classification of maps Between pseudo- projective spaces,Questiones Mathematicae,11(1988),409-422.
[23] J.W.Rutter,The groups of homotopy self equivalences classes of CW- complexes, Math. Proc. Camb. Phil. Soc,93 (1983),275-293.
[24] J.W.Rutter,Groups of self-homotopy equivalences of induced space,Comm. Math. Helv,45(1970),236-255.
[25] J.W.Rutter,Homotopy self-equivalences 1988-1999.In Groups of homoopy self-equivalences and related topics,(ed. Ken-ichi Maruyama and John W.Rutter),Contemporary Mathematics 274(2001),1-11.
[26] N.Sawashita, On the group of self-equivalences of the product of spheres. Hiroshima Math. J.5(1975),69-86.
[27] W.Shin,On the group ξ[X] of homotopy equivalence maps. Bull. Amer. Math. Soc.72 (1964),361-365.
[28] A.J.Sieradske,Twisted self-homotopy equivalences,Pacific J.Math.3 (1970),789-802.
[29] E.H.Spanier,Algebraic Topology,(Springer, 1966).
[30] G.W.Whitehead,Elements of homotopy theory, GTM 61 (Springer,1978).
[31] T.Yamanoshita,On the spaces of self-homotopy equivalences of certain CW-complexes,J.Math. Soc. Jpn 37(1985),455-470.
[32] K.Yamaguchi,On the self-homotopy of wedge of certain complexed,Kodai Math. J,6(1983), 1-30.