Remark 11.4.4. A \(\mathrm {Spin}^c\)-structure on \(M\) determines a \(\KU \)-orientation of \([T_M]\) through the construction of Atiyah et al. (1964). At the other extreme, Example 10.4.9 shows that every real vector bundle is canonically \(\MO \)-orientable. This ties generalized Poincaré duality directly to the bordism theory of Section 10.3.
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