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Is it true that the Cartesian product of two spin manifolds is spin?

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    $\begingroup$ Yes, for example see Milnor's cute paper "Remarks concerning spin manifolds". In particular using the Whitney sum this implies the "spin bordism ring" is a ring. $\endgroup$
    – Chris Gerig
    Commented Aug 24, 2022 at 16:00
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    $\begingroup$ This amounts to the existence of a homomorphism $\text{Spin}(n) \times \text{Spin}(m) \to \text{Spin}(n+m)$ lifting the direct sum homomorphism between orthogonal groups, which is quickly constructed in terms of whatever your favorite definition of Spin is. With the double cover perspective, special care must be taken with Spin(1) and Spin(2). With the Clifford algebra perspective there is no difficulty. $\endgroup$
    – mme
    Commented Aug 24, 2022 at 16:18

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