Detecting framed manifolds in the 8 and 16 stems

Abstract

In this talk we wish to apply some of the ideas developed in [3]. Our philosophy is to try and investigate the stable stems using the geometry of framed manifolds, and without recourse to homological algebra. Our chief tool is the e invariant in symplectic cobordismo We are able to show that both the 8 and 16 stems, S 8 and S16 , contain a subgroup 7~2~7L2, where in each case one summand is in the image of J (and consequently, the other is not). We give explicit framed manifolds for each element. The above consequences flow at once from our main result, which is Table (4.4). Of course, an illicit peep at Toda reveals that we have in fact captured the whole of the 8 and 16 stems in this fashion. In other words our e invariant is, in theory, perfect. However, we know of no way to establish this pleasant fact using our methods alone. For those who cannot live without an Ext, we remark that (4.4) can be rephrased to read