Question

Let $X$ be a compact oriented manifold, and $A$ and $B$ closed oriented submanifolds intersecting cleanly. Then I've always been under the impression that pushing forward a cohomology class from $A$ to $X$ and then pulling back from $B$ should have a base change formula where instead one pulls back to $A\cap B$ and pushes forward to $B$.

Of course, this couldn't possibly be right if $A$ and $B$ aren't transverse. I think in the non-transverse case, one should correct by the Euler class of the excess bundle $T_{A\cap B}X/(T_{A\cap B}A+T_{A\cap B}B)$.

All of my intuition for algebraic topology tells that this true and easy to prove, but of course, one can't write that in a paper.

Does anyone know a convenient reference for this fact? I've tried to find it via Google, but apparently can't find the right keywords, and a quick scan of Hatcher came up negative.

Answer 1

Apologies if this is too late, but the canonical reference for this is Quillen's seminal paper "Elementary proofs of some results of cobordism theory using Steenrod operations" Advances in Math. 7 1971 29--56 (1971). The proof given there is for complex cobordism and is entirely geometric. Presumably Quillen learned this from Bott, who gives a clean intersection formula in his paper "On the iteration of closed geodesics and the Sturm intersection theory" Comm. Pure Appl. Math. 9 (1956), 171--206. If you are interested in the generalisation to immersions, see F. Ronga, "On multiple points of smooth immersions" Comment. Math. Helv. 55 (1980), no. 4, 521--527.