On the homotopy type of the space of generalized Morse functions

LET N BE A compact smooth (Cm) n-manifold and let g: N--, R be a fixed Morse function without critical points on 8N. We shall say that a functionf: N + R is admissible if it is smooth and agrees with g near 3N. Let 9(N, 8N) denote the space of all admissible functions on N. Then F(N, 8N) is convex and thus contractible. (We use the strong C”-topology[2] for all function spaces.) Let A(N, 8N) denote the space of all admissible Morse functions on N. Then A(N, 8N) has many components since functions in a single component have the same number of critical points. To obtain a connected space of functions we must allow cancellation of critical points. Thus let &‘(N, 8N) denote the space of all admissible functions of N whose critical points are either Morse or birth-death singularities (defined in 2.1). In this paper we compute the (n I)-homotopy type of S’(N, dN). Let f: N + IF! be an admissible function. Then the derivative of f gives a map N/aN + T(r) where T(t) is the Thorn space of the tangent bundle r of N. This map is given as follows. We choose a fixed Riemannian metric on N by embedding N in R” and we let E = min (1, inf { IlVg(x)llIx E aN}). Then the composition of l/~Vfz N -+ E with the collapsing map E + T(T) where E is the total space of T sends aN to *, so it induces a map N/aN + T(r). Suppose that f E A(N, 8N). Then we can choose E > 0 a continuous function off so that U = {x E NI/Vf(x)II < L } c int N = N 8N and U is the union of disjoint contractible neighborhoods Vi of the critical points yi off. Using this E we again get a map N/aN + T(z). But this map now lifts to a map N/aN + T@ *r ) where p *z is the pull-back of r along the projection p: BO x N-P N. The lifting is given by sending x E Vi to (Pi, l/cvf(x)) where Pi is the plane in IF” spanned by the negative eigenvectors of P~(JJJ considered as an element of BO. (The complement of U goes to *.) In the special case when N is embedded in R” this gives an element of Q”P((BO x N),) where the lower “ + ” means add a disjoint base point. In other cases we get an element of CPS”((B0 x N),) by passing to the normal disk bundle of N by a construction called suspension (3.1.a). In any case we get a map J/: .&(N, cYN)+CPS~((BO x N),). In $5 we show how the negative eigenplanes of cancelling Morse singularities can be made to match up at a birth-death singularity thus allowing us to extend the function I,+ to H(N, 8N). The composition &‘(N, dN)42”S”((BO x N)+)-+fPSm(N+) is null homotopic since it extends to all of P(N, 8N). Consequently we get a map 6: H(N, aN)+ W’Soc(BO A (N,)) N fiber [QmSm((BO x N)+)--,R”S”(N+)]. The main theorem of this paper is that this map is n-connected. The (n l)-connectivity of $ is due to the author. This was extended to n-connectivity by Crichton Ogle in $4. The main element in the proof is our result in [3] where we establish conditions for the elimination of singularities other than Morse and birth-death singularities from a parameterized family of admissible functions on N. In $6 we show that spaces of functions with restricted Thorn-Boardman singularities look like QmS5(I+’ A (N,)). More precisely for any Thorn-Boardman index