A proof of the nonretractibility of a cell onto its boundary

SHORTER NOTES The purpose of this department is to publish very short papers of an unusually elegant and polished character, for which there is nor- mally no other outlet. A PROOF OF THE NONRETRACTIBILITY OF A CELL ONTO ITS BOUNDARY MORRIS W. HIRSCH By appealing to the simplicial approximation theorem [2, p. 64] it suffices to prove that there is no simplicial retraction of a subdivi- sion of a closed ra-simplex E onto its boundary dE. Suppose/: £—>d£ is a simplicial retraction. Let a be the barycenter of an (n— l)-simplex A EdE. The point is this:/_1(a) is a compact one- dimensional manifold whose boundary is contained in dE. The com- ponent of /_1(a) containing a is thus a broken line segment with one endpoint at a; but the other endpoint cannot exist. It would have to be a point of dE different from a which maps onto a under /, contra- dicting the assumption that/| dE is the identity. The proof that/_1(o) has the stated properties is simple and classi- cal (cf. [3]). Any ra-simplex B mapping onto A has precisely two faces mapping onto A, so that BC\f^1(a) is the line segment in B joining the barycenters of the two faces. These line segments fit together to form a manifold whose boundary is in dE because every (n— 1)- simplex C of E is incident to either one or two ra-simplices, according to whether CEdE or not. The same proof applies if £ is a compact triangulated manifold with boundary dE. More generally, the proof works if £ is a finite ra-dimensional complex such that each (ra— 1)-simplex is a face of not more than two ra-simplices, and dE is the union of those (ra—1)- simplices incident to at most one w-simplex. In the case where £ is a compact differentiable manifold, one may use the differentiable approximation theorem in place of the sim- plicial one, and take a to be a regular value. This of course requires a theorem such as Brown's [l, Theorem 3.Ill], Dubovitski's Theorem 4], or Sard's [5] on the existence of regular values. References 1. A. B. Brown, Functional dependence, Trans. Amer. Math. Received by the editors October 8, 1962. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Soc. 38 (1935)