V. Voevodsky [6] invented the category of framed correspondences with the hope to give a new construction of stable motivic homotopy theory SH(k) which will be more friendly for computational purposes. Joint with G. Garkusha we used framed correspondences to develop the theory of framed motives in [4]. This theory led us in [5] to a genuinely local construction of SH(k). In particular, we get rid of motivic equivalences completely.
In my lectures I will recall the definition of framed correspondences and describe the genuinely local model for SH(k) (assuming that the base field k is infinite and perfect). I will also discuss several applications. Let Fr(Y,X) be the pointed set of stable framed correspondences between smooth algebraic varieties Y and X. For the first two applications I choose k = ℂ for simplicity. For further two applications k is any infinite and perfect field.
(1) The simplicial space Fr(