Compact Abelian Transformation Groups
L. N. Mann · Transactions of the American Mathematical Society · 1961
The special homology theory we present now is due to Smith for % -Zp.Let K be a finite simplicial complex and T, a periodic simplicial map on K oi period p.Let M be a subcomplex of K invariant under T (i.e., T(M) = M).We denote the fixed point set of T by L and assume that L and MC\L are subcomplexes of K and M respectively.We moreover assume that T is a primitive simplicial map.By this we mean that each simplex in K -L has p distinct Gimages.We let C¡(K), ZS(K), Bj(K) and H¡(K) denote the group of j-chains, j-cycles, bounding j'-cycles and the j-homology group of K with coefficient group g.The boundary operator is denoted by d.Now let r denote the chain map 1 -T of C¡(K) into itself ander the chain map: 1 + T+T2+ ■ ■ • + TP~1.Clearly ot = 0 = to\ We use p or p to denote either a or t (i.e. if p=cr, p=r and vice versa).We may now define the special homology groups Hfi(K, M) and HP,(K, M) with respect to g as follows:(Let C*(K, M)= E;-oGCK, M), C%(K, M) = 237-0 CJ(K, M) and CILK, M) = 237-0 Q(K, M).We shall now define a map 7: G(f, M) ->L^(2C, M).Let (c+C*(M))QC*(K, M).We Iet7(c + G(M)) = (pc + Cl(M)) QC*(K, M).It is easy to see that y is well defined and a homomorphism onto.Let /3 denote the inclusion map ß: C%(K, M)-^>C*(K, M).We have then the exact sequence (2.1) 0 -> Cl(K, M) t C*(K, M) ^ cl(K, M) -» 0.Since ß and y commute with d (since d T-Td), (2.1) gives rise to the following exact sequence:(2.2)-► tfm(K, M) ^ H°i(K, M) -^ H^K, M) A H*i(K, M) -* • • • where /3* and y are induced by ß and y and a* is an induced boundary homomorphism.