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I am interested in figuring out the following submanifold generator of a $Pin^+$ Bordism or Cobordism group in the dimension 4, say for a $Pin^+$ cobordism group of the classifying space $BG=BSU(2)$:

$$\Omega^{4,\text{Pin}^{+} }_{\text{tors}}(BSU(2)_{\text{Spin}},U(1))= \text{Hom}(\Omega^{\text{Pin}^{+}}_{4,\text{tor}}(BSU(2)),U(1))$$

The answer is an Abelian group, and it should be a cyclic group of order 4 or of order 8. My question is what are the submanifold generator of this cobordism group? i.e. the submanifold generator of this Pontryagin-dual of the torsion subgroup of the $Pin^+$ bordism group (dimension 4th)?

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The 4-dimensional $\operatorname{Pin}^+$ bordism group is $\mathbb{Z}/16$, generated by $\mathbb{RP}^4$. The reference is an old paper of Kirby and Taylor, Pin structures on low dimensional manifolds, published in the LMS Lecture Notes 151, edited by Donaldson and Thomas, pages 177-242. See page 215, Theorem 5.2

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  • $\begingroup$ I also wish to know the result for $SU(2)$. $\endgroup$
    – annie marie cœur
    Commented Jun 18, 2017 at 0:50

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