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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.

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I think these matters are discussed (in much greater generality) in Dold's "Lectures on algebraic topology" (Chapter VIII, Section 13), which can be partially read at google.books. – Igor Belegradek Apr 28 '10 at 23:19
up vote 3 down vote accepted

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.

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It's too late in the sense that the paper is already published, but I still appreciate the answer. – Ben Webster Aug 31 '10 at 17:16

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