特殊多面体的笵形和同伦群

<正> 多面体的伦型鉴定问题,已经有过许多拓扑工作者的研究.J.H.C.Whitehead找到 A_n~2多面体的伦型和正则上同调环头(n=2)或正则上同调系统类(n>2)间的一一对应.他和 S.Machane 发见“三型”所对应的代数构造,于是引起(?)

ON NORMAL FORMS OF HOMOTOPY TYPE AND HOMOTOPY GROUPS OF CERTAIN POLYHEDRA

The homotopy group,∏_n(x),(n≥2)of a given arcwise connected polyhedronis an abelian group,hence may be considered as a direct sum,∑ Ζ_(p~r),of cyclicgroups of prime power order.As a consequence we have a function,φ_n,definedon powers of primes such that φ_n(p~r)is the number of copies of Ζ_(p~r)occurringin H_n(x)=∑ Ζ_(p~r).No doubt φ_n(p~r)is an invariant of homotopy type.Thoughφ_n(p~r)is theoretically determined if the space is given,but up to this time there is still no definite method to produce such invariants.Since there is atopological space whose homotopy groups may be arbitrary assigned,there mustbe a great deal of integer valued invariants of homotopy type.This papercontains detail exposition of results announced in two short notes published inScience Record.By simple A_n~3-polyhedra we mean those whose cohomology groupsH~(n+r)(r=1,2,3)may be written as a direct sum H_1~(n+r)+H_2~(n+r),where H_1~(n+r)is the direct sum of q_r(≥0)cyclic groups of the same order,2~(p_r),and H_2~(n+r)isthe direct sum of a free groups(if any)and cyclic groups of odd prime powerorders(if any).As a result of theory of proper isomorphisms block invariants,relative block invariants,characteristic polynomials,characteristic coefficients,Φ_1-torsions and Φ_2-torsions are introduced to constitute a complete and inde-pendent system of integer valued homotopy invariants of simple A_n~3-polyhedra(n>3)besides Betti numbers and torsions.To each Of the above invariants,τ,say,there is a normal simple A_n~3-polyhedron, N_τ.If χ is a simple A_n~3-poly-hedron,then it is proved that∏_8(x)=∑_τ∏_s(N_τ),if s<2n-1,(1)where ∑ denotes direct sum and τ∏_r(N_τ)is a direct sum ∏_r(N_τ)+…+∏_r(N_τ)ofτ copies of ∏_r(N_τ).This shows relations between φ_s(p~r)(x)and τ's if s<2n-1.In other words,φ_s(p~r)(x)is expressed as a linear expression of τ's with co-efficients φ_s(p~r)(N_τ)if s<2n-1.In case s≥2n-1,then in φ_s(p~r)quadraticforms of τ's or other terms are expected to enter.These phenomena are stillobscure.The formula(1)shows the importance to compute homotopy groups of∏_s(N_τ).But ∏_(n+2)(N_τ)are easily computed for all τ.The method used here is also effective for general A_n~3-polyhedra if n>3.When n=3,we must classify A_n~3-cohomology rings under proper automorphismspreserving the normal forms of γ,γ_(n+1),Φ_1 and Φ_2.This needs further study infuture.FOOTNOTES1)In case H_r(x)(r=1,…)are of finite type,then φ_n(p~r)is finite fordefinite n,r and p.See20].2)To a characteristic polynomial and one of its corresponding characteristiccoefficient correspond one invariant and one normal polyhedron.3)See9]and14].