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A homotopy class of axial maps pn X p _ pp+k determines an invariant in Tn(Vn+k+l,n+l) (2k > n + 2). If an axial map is symmetric and has trivial invariant it extends to a symmetric axial map pn+l x pn+l _ pn+k+I, An immersion of pn in Rn+k lifts to an immersion of Sn in Rn+k and so has a Smale invariant. For j: Rn+k -+ Rn+k+2, 2k > n + 2 (resp. 2k > n + 3), any embedding a: pn -+ Rn+k with trivial Smale invariant induces an embedding of pn+l in Rn+k+2 whose restriction to pn is regularly homotopic (resp. isotopic) to ja. Let pn denote real projective n-space. A map P' x pn > pnl+k is axial if it is homotopic to the inclusion pn fl pn+k when composed with either inclusion P'n -* X p n -* p n X pn, pn -pnX * -* p n X p n, and symmetric axial if also it is equivariant with respect to the interchange involution on Pn X pn and the trivial involution on pn+k. This note defines an invariant of the homotopy class of such a map and examines the consequences of its vanishing. Let f: Sn X Sn -> Sn+k have f(x, -y) = -f(x,y). Then exponential correspondence yields a map f': S n , Yn+k+ln+l' where Y is the space of fo SSIto SrI ir,s antipodal-involution-preserving maps from S introduced in [5, ? 1]. For 2k > n + 2, f' lifts by [5, (1.1)] to a map into the Stiefel manifold Vn+k+l,n+l, whose homotopy class of E 7n(V+k+ln+I)is an invariant of the equivariant homotopy class of f. If f covers an axial map f: pnlX pn > pn+k we write ck equally for kj LEMMA 1. Suppose 2k > n + 2. Let f be a symmetric axial map pn X pn p n+k (whose invariant A E s7n (+k+l,n+l) is the trivial element. Then i o f extends to a symmetric axial map g: pn+l> pn+l >pn+k+l(i:pn+k C.. pn+k+l). Further, if symmetric axial f: PnX pn J -l> pn+k is homotopic to J, then i o J' extends to g' homotopic to g; if j' is moreover symmetric homotopic to f (i.e. via a symmetry-preserving homotopy), then g' is symmetric homotopic to g. PROOF. Let f: Sn X 5n >5n+k coverf. The condition ensures thatf may be extended to fi : CSn X Sn -> 5n+k, where CSn = (SnX I)/(SnX {l}). Make the identification SN+l CSN U (SN X [o, _ 1])/(SN X {-l}). Then f1 exReceived by the editors October 13, 1975. AMS (MOS) subject classifications (1970). Primary 55E05, 57D40.

Induction on symmetric axial maps and embeddings of projective spaces