TITLE: Reference for base change of cohomology pull-push for clean intersections.
QUESTION [6 upvotes]: 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.

REPLY [5 votes]: 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.