On Chains of Topological Spaces

Abstract

complex. By a chain-complex L of T we shall mean a subcomplex of R. Meshes, chain-deformations, e or otherwise, said to be on 9?, are then defined in the natural way. The complex L is thus a chain-complex in the sense of No. 2, except that the (1)-chains and Kronecker-indices may now be elements of (M and not merely integers. The q-chains of L, q > 0, are still assumed with integer coefficients. The group 3 enters in them indirectly through the components Cq of the chains. 8. Chain-groups. Our first difficulties arise with the definition of addition. We shall find it convenient in fact to distinguish two kinds of additions, the one of geometric type, the other more algebraic. Suppose that we have two chains Cq = { Cq}, cq = {Cq}, associated with the same projection-sequence {K'}. Then {Cq + C.') is another chain with the same projection-sequence. We call it geometric sum of cq, cq and designate it by Cq + cq in the usual manner. It is also immediately apparent that (8.1) -(Cq + Cq) = (-Cq) + (-Cq), (8.2) F(Cq + Cq) = F(Cq) + F(c'). The definition of addition just given is the most natural. It suffers however from the grave defect that the sum Cq + cq is only defined for certain special pairs Cq, Cq and hence it may not be used to form an additive group comprising all the q-chains as elements. We must therefore look for a somewhat different addition. It will be in fact an addition of chain-classes which are based on a certain notion of equivalence to be developed presently. 9. Let {Ki}, be two projection-sequences with projections 7ri, 7r, and let there exist, for each i, a deformation Oi:Kt -* K't whose amount -i3 0 with 1/i, and where in addition Oi7ri = 7r' Oi+ 1. We call the aggregate 0 = { iO a regular deformation of the first projection-sequence into the second, a regular isotopy whenever every Oi is isotopic. In the latter case the inverses OH 1 are defined and isotopic and 10 1} defines a regular isotopy, the inverse of 0 and denoted by O-1. Similarly if 0' = { l'o is a second regular deformation , {K'}, {O' OJ} defines a regular deformation {Kt} -* {K'} the product of the two and denoted by 0'0. Finally if every Oi = 1, we write 0 = 1. Let now Cq be the same chain as previously, attached to {Kt}, and let 0 be a regular deformation {Kt} -* {K'tK. If Cqi = OiCq, we have Cq = Oi7rid+ = ir$C+', so that c' = {Ci} is a E-chain attached to {K'I}. We shall denote this new chain by OCq. We notice that under our conventions Oig = 9, so that Oc-1 = c-1. Moreover since OYF = Fi, 6i(C') = OiCi, we have OF = FO, O(-c) = Oc, and for q = 0, 0(co) = (co). Finally since Oi is an ei deformation, where ei -f 0 with 1/i, we have I OCq = Cq 1, or 0 does not modify the sets associated with the E-chains. This content downloaded from 207.46.13.105 on Wed, 25 May 2016 06:46:00 UTC All use subject to http://about.jstor.org/terms