Embedding homotopy cells

Abstract

PROOF. Let B = Bd (Wk X 16-k) and A1 and A2 be two copies of WkXI1-k. Also let B1=Bd (WkTXXIS-k). Then B=APUA2, where A1CA2=Bl. We shall now wish to calculate ri(B) and w2(B). Now w7l(Wk XIS-k) =w7r(Wk) = 1 and A1nA2 is connected. Hence by Van Kampen's theorem wr,(B) =0. By Hurewicz's theorem H2(B) = r2(B). Again H2(Wk X IS-k) = H2(Wk) and from the Mayer-Vietoris sequence of the proper triad (B, A1, A2) we see that 72(B) =H2(B) =0. It follows from [3] that B = S5. Now