The answer is a paper aptly titled Canonical framings for 3-manifolds by Rob Kirby and Paul Melvin.

Here is a 2012 email correspondence between me and Kirby concerning that paper (glad I didn't walk next door to ask him otherwise I may have forgotten):

• Chris: For a spin 3-manifold M, the spin structure s+s on TM+TM is independent of the choice of spin structure s on TM. Could you explain why?
• Kirby: Here is a down to earth way to think about it. A spin structure is a trivialization over the 1-skeleton which extends (you don't care how) over the 2-skeleton. There are two ways to trivialize over a circle, corresponding to \pi_1 of S(n) n>2, and in the case of n=2 or 1 we work mod 2. Then on s+s, if you change the trivialization of s over a circle, then you double that change for s+s, which mod 0 is no change.

The answer is a paper aptly titled Canonical framings for 3-manifolds by Rob Kirby and Paul Melvin.

Here is a 2012 email correspondence between me and Kirby concerning that paper (glad I didn't walk over to his office otherwise I may have forgotten verbally):

• Chris: For a spin 3-manifold M, the spin structure s+s on TM+TM is independent of the choice of spin structure s on TM. Could you explain why?
• Kirby: Here is a down to earth way to think about it. A spin structure is a trivialization over the 1-skeleton which extends (you don't care how) over the 2-skeleton. There are two ways to trivialize over a circle, corresponding to \pi_1 of S(n) n>2, and in the case of n=2 or 1 we work mod 2. Then on s+s, if you change the trivialization of s over a circle, then you double that change for s+s, which mod 0 is no change.

The answer is a paper aptly titled Canonical framings for 3-manifolds by Rob Kirby and Paul Melvin.

In case it becomes relevant, here is a 2012 email between me and Kirby concerning that paper:

• Chris: For a spin 3-manifold M, the spin structure s+s on TM+TM is independent of the choice of spin structure s on TM. Could you explain why?
• Kirby: Here is a down to earth way to think about it. A spin structure is a trivialization over the 1-skeleton which extends (you don't care how) over the 2-skeleton. There are two ways to trivialize over a circle, corresponding to \pi_1 of S(n) n>2, and in the case of n=2 or 1 we work mod 2. Then on s+s, if you change the trivialization of s over a circle, then you double that change for s+s, which mod 0 is no change.

The answer is a paper aptly titled Canonical framings for 3-manifolds by Rob Kirby and Paul Melvin.

Here is a 2012 email correspondence between me and Kirby concerning that paper (glad I didn't walk next door to ask him otherwise I may have forgotten):

• Chris: For a spin 3-manifold M, the spin structure s+s on TM+TM is independent of the choice of spin structure s on TM. Could you explain why?
• Kirby: Here is a down to earth way to think about it. A spin structure is a trivialization over the 1-skeleton which extends (you don't care how) over the 2-skeleton. There are two ways to trivialize over a circle, corresponding to \pi_1 of S(n) n>2, and in the case of n=2 or 1 we work mod 2. Then on s+s, if you change the trivialization of s over a circle, then you double that change for s+s, which mod 0 is no change.