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attaching map basepoint c.r.p. of type c.r.p. on Sn cF(f classes of maps consider the maps constant map Corollary Ed(F ET+1 exists a map exists a s.c.r.p. explicit form fixed map following properties form without brackets Hence homotopic to g homotopy classes HOMOTOPY THEORY Hopf invariant Hopf suspension Hp+q(L induction integers Ip+q isomorphic Lemma Let Let tn map F map g map of type maps Sn maps SnxSn maps which agree mn(Sn Mp,q nn(Sn nn+i(Sn notations np+q(X np+q+2(Sr+2 nr+i(X obtain product on Sn PRODUCTS ON SPHERES Proof prove Q.E.D. Remark quaternionic radial projection reflecting product relative homology groups right suspension rotation group separation element simplicial complex Sn be maps Sn of type Sn x Sn SnxSn Sn Sr+1 which fulfil subspace Suppose SxSey theorem 6.2 Theorem Let tn+i v(ln x a WAALRE Whitehead product Xp+q Xp+q+i Yx[SxSeye zijn