RECENT ADVANCES INHOMOTOPY THEORY 5 are homomorphisms* (L17) The two natural group structures on [SX, £IY] coincide and are abelian, (1.18) If n2, then [S n X, Y ] and [X, Q n Y] are abelian. The suspensio n an d loo p functor s preserv e homotop y an d therefor e defin e map s 5 , : [X, Y] -* [SX, SY], Qm:[X, Y]-+[QX,QY], and (1.19) The maps Sm: [SX, Y]~*[S 2 X,SY], a*:[x, QY]-+[nx 9 n2Y], are homomorphisms. Let \X, Y\# b e th e grade d abelia n grou p fo r whic h \X 9 Y\ i s th e direc t limi t Km[SQ+kX, S k Y] k under th e homomorphism s S,: [S q +kX, S k Y] ~* [S q+k +lX, S k+l Y]. The element s o f {X, Y\ ar e calle d S°maps of degree q. As a specia l case , w e hav e th e stable homotopy an d cohomotopy groups oq(X) = {S°, Xi q , o*(X) = \X,S°\_ q . In particular , le t a = a (S° ) = iS°, S°! = {S«, S°\ = a'" q q q be th e stable group of the q-stem 0 Let fz X — Y b e a ma p i n S „ The mapping cylinder I, i s th e spac e obtaine d from (X A / ) V y b y identifyin g x A 1 wit h f(x) fo r al l x € X, Similarly , th e map- ping cone T f i s obtaine d fro m (X A T) V Y b y makin g th e sam e identifications . Th e map x — x A 0 imbed s I a s a subspac e o f / . Similarl y th e inclusion s o f Y i n (X A 1) V Y an d (Z A D V y induc e imbedding s o f Y i n / , an d T f respectively . Moreover,

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