Answer by Mohammed Abouzaid for Parametrized Atiyah-Singer index theorem
Since Thomas is satisfied with my comment, I will post it as an answer to close: this result is in the 4th Atiyah-Singer paper: jstor.org/stable/1970756
View ArticleParametrized Atiyah-Singer index theorem
Let $M$ be any smooth manifold (could be unorientable - I think). Let $E,F \to M$ be two complex vector bundles. Let $S$ be any compact space, and let $D_s:E\to F,s\in S$ be a continuous family...
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