Embeddings in the 3/4 range

We give a complete obstruction to turning an immersion f : M m → R n into an embedding when 3 n ⩾ 4 m + 5 . It is a secondary obstruction, and exists only when the primary obstruction, due to André Haefliger, vanishes. The obstruction lives in a twisted cobordism group, and its vanishing implies the existence of an embedding in the regular homotopy class of f in the range indicated. We use Tom Goodwillieʹs calculus of functors, following Michael Weiss, to help organize and prove the result.