I was just reading This question Wherein the OP wants to know what the tangent bundle looks like in a general connected sum $$T\left(M_{1}\#M_{2}\right)$$ (where the connected sum is along some submanifold $V$). The only answer provided (first comment) is that at points interior to $M_{i}$ with $V$ removed, the tangent bundle looks like $T(M_{i})$.

I'm interested in the analog for spin bundles on a connected sum. In general, a spin manifold might be the connected sum of several non-spin manifolds.

The previous comment won't apply in this case since there is no spinor bundle for some $M_i $. In that case how do we find an isomorphism that could apply to our boundary space?

I had a hunch that might involve generalized spin structures $spin_G $ which I've also asked a question about. If anyone could maybe clarify how this might work I'd be very thankful!

To be more specific, I am particularly interested in the induced spin structure on $V$. It would seem like one orientation would have one form and the opposite orientation would have another? My thought was there must be some kind of spin statistics requirement on the boundary of the connected manifolds.

Consider for example the connected sum of the complex projective plane and say $S^3 \times S^1 $. The former admits a $Spin_c $ structure while the latter admits a standard spin structure, how to they meet and agree on $V$ ?



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