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I will phrase this question in terms of attaching smooth manifolds along a submanifold, though it is certainly more general.

Let $M_1$ and $M_2$ be smooth $n$-manifolds (maybe closed, for simplicity), $N$ a closed $k$-manifold, $D$ a closed $(n-k)$-disk bundle over $N$ (so that $D$ is an $n$-dimensional manifold whose boundary is the associated $(n-k-1)$-sphere bundle), and suppose we have embeddings $f_i:D\rightarrow M_i$.

Now say $\tilde{M_i}=M_i\setminus f(N)$, let $E_0$ be the punctured-open-disk bundle associated with $D$, and let $\alpha:E_0\rightarrow E_0$ be defined by sending $v_x$ in the fiber of $x\in N$ to the point $(1-|v_x|)\frac{v_x}{|v_x|}$ in the same fiber (intuitively, $\alpha$ turns $E_0$ inside-out so that we can attach the manifolds with a "collar").

Then if we form the Topological pushout of $\tilde{M_1}$ and $\tilde{M_2}$ using the smooth embeddings $f_1|_{E_0}$ and $f_2|_{E_0}\circ \alpha$, this in fact produces a pushout in the smooth category. The resulting manifold $M$ then has a tangent bundle, and in fact this tangent bundle can be formed by attaching the tangent bundles of $M_1$ and $M_2$ using the same recipe.

So, finally, here is the question: is there a formula for characteristic classes of $M$ (maybe just restrict attention to Stiefel-Whitney, Chern, Pontryagin classes) in terms of the characteristic classes of $M_1$, $M_2$, and $N$ (and the embeddings $f_1$, $f_2$)? More generally, is there a similar formula for attaching arbitrary bundles over arbitrary topological spaces (i.e. we would form a topological pushout on the base spaces and indicate how the fibers would be identified over points that are glued together)?

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If you want a formula you're going to have to go up to twisted coefficients, I think. For example, consider constructing a Moebius band as the union of two discs, glued together along two arcs in their boundaries. But with Stiefel-Whitney classes, there's no way of comparing the extensions so I don't think there's a formula. – Ryan Budney Feb 8 '12 at 0:09
Said another way, if you only look at the Stiefel-Whitney classes of $M_1, M_2$ and $N$ and the maps between them, there's no way of telling the Moebius band apart from a cylinder, when you write it as a union of two discs. – Ryan Budney Feb 8 '12 at 0:26
Hey you. Have you managed to find a formula for the SW-classes of a connected sum $M_1\sharp M_2$ in terms of those of $M_1$, $M_2$ (the case when $N$ is a point)? This might be a good place to start. – Mark Grant Feb 8 '12 at 13:05
up vote 3 down vote accepted

It depends what you mean by "formula", since you are talking about cohomology classes in different manifolds, so at the very least you need a way to relate them, which depends on context. So the "answer" is the Mayer-Vietoris sequence. Naturality of characteristic classes, together with the fact that the characteristic classes of (the tangent bundle of) a disk bundle $D(E)->M$ can be computed using $TD=TM\oplus E$ gives you some information which you then have to put together.

For characteristic numbers, there are formulas, eg Novikov additivity gives a formula for the signature of $M$ in terms of those of $M_1, M_2$ and $D$. Similarly for the Euler characteristic. If you glue $D\times I$ to $(M_1\cup M_2)\times I$ you get a bordism from $M_1\cup M_2$ to $M$ and so bordism invariance of characteristic numbers can be helpful.

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