Note on the Homotopy Type of Mapping Cones"
PETER HILTON
I n [2] and [3] we proved certain theorems about the semi-group of homotopy types of based spaces where the composition operation is just the disjoint union with identification of base points. These results were suggested by certain observations of Freyd [l], which led him to deep theorems of stable homotopy theory, but they are not themselves difficult to prove. In the generality adopted in [3] we have the following results, where A , B are arbitrary based spaces, C is the suspension functor, and C, is the mapping conel of the homotopy class u.
Let u E (CA, B ) be an element of order k, let I be prime to k and let / 3 = Lu.
Then we have proved THEOREM 1. (i) C, + C2A N C, + P A . (ii) I f A and u are suspensions, then C, + B N Ca + B.
(iii) V A is a suspension, then t C , cv tC, provided It = I)I.1 (mod k) ; in particular, where 4 is Euler's function. 4 4 W a 'u W C , 3
In this note we obtain further relations between the homotopy types C, ; these relations were again suggested by some (unpublished) results of Freyd. We use two basic lemmas; the first is well-known. LEMMA 2. Let XI % X Y 3 Yl , where u, v are homotopy equivalences. Then c, N Curv * Now given spaces X I , X, , Yl , Y, , of which XI, X, are suspensions, let us * This article represents the results of research carried out at the Courant Institute, New York University, with the support of the Office of Naval Research, Contract No. None 285(46) and the National Science Foundation, grant NSF GP 6357. Reproduction in whole or in part is permitted for any purpose of t h e United States Government.
Thus C , is the homotopy type of the mapping cone of a m a p f i n the class a. We take this opportunity to point out an error in [3] ; the hypotheses of Theorem 1.6(b) and (c) should include: "If A is a suspension." The hypothesis that a is a suspension is quite superfluous in (1.3) and Theorem 1.6(c).