Abstract
This paper aims to give a combinatorial characterization and also construct representations of the fundamental groups of the submanifolds on flat Robertson-Walker space by using some geometrical transformations. The homotopy groups of the limit folding on flat Robertson-Walker space are presented. The homotopy groups of the retractions and deformation retract on flat Robertson-Walker space are obtained. The fundamental groups of some types of geodesics in the flat Robertson-Walker space are discussed. New types of homotopy maps are deduced. Theorems governing this connection are achieved.
Highlights
Introduction andDefinitionsRobertson-Walker space represents one of the most intriguing and emblematic discoveries in the history of geometry
An operator which assigns to every object in one category a corresponding object in another category and to every map in the first a map in the second in such a way that compositions are preserved and the identity map is taken to the identity map is called a functor
We may summarize our activities far by saying that we have constructed a functor from the category of pointed spaces and maps to the category of groups and homomorphisms. Such functors are the vehicles by which one translates topological problems into algebraic problem [8, 10, 12, 22,23,24,25]
Summary
Robertson-Walker space represents one of the most intriguing and emblematic discoveries in the history of geometry. { } are given by x1 = t, x2 = - a(t)r sinq, x3 =0, x4 =a(t)r cosq , which is a hypersurface F6 Ã w4 in flat Robertson Walker space which is a geodesic and a retraction. The fundamental group of the deformation retract for the folded flat Robertson -Walker space π1 (h¡ {w4 – (μi)}) onto the folded geodesic π1 (h¡ (S1)) is a (t)r sinq sinf, atPh(2etÃ)pfrl1a{c(pwtoh1Rs4(q–moh}b¡(-,eμc(r()im)mt)}s,i=oi)csn})ln)+g-=ieWlv(n1e(-aen-hlh)k{{b2e0{y2crt,,0+sa,p20(at),)car{e(s{ti{tn)w,rqa}4.c(–tTo)(shrμfusi),is}na, oqp(ntc1)tor{oswsaif4ng–q,easo((iμtdn)i)efr}s,si≈ican(qt)sπri1cno(fhsp,qa¡1}((-th{)w¡(rm4c(m–io)s}(,qμ+c)i})l)n-}=)e(≈ha({)-0π},12+0(2ch,c0+¡{,02a(,)(Sat{1)(){rt)t}),risasi(istno)rqms,0ion,raqp(chto)icrs f ,a (t)r sinq cos q} to Z. sinf,a (t)r c p1 (P2 Ã {w4 – (μi)}) is isomorphic to identity group. The fundamental group of the deformation retract for the folded flat Robertson -Walker space π1 (h¡ {w4 – (μi)}) onto the folded geodesic π1 (h¡ (F5)) is. Let X Ã {W2 – (μi)} be the union of the circles
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