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If $E \to X$ is a complex vector bundle, its Thom class is defined using the exterior algebra of $E$, giving the Thom isomorphism $K^*(X) \cong \widetilde{K^*}(X^E)$. Atiyah-Bott-Shapiro use Clifford modules to construct the Thom class of an even dimensional real vector bundle with $Spin^c$-structure. Note that the Thom isomorphism in both cases does not shift the degree of K-theory.

In Atiyah and Hirzebruch's paper Vector bundles and homogeneous spaces, page 210, they state that their exists a pushforward for '$c_1$-maps' $f \colon X \to Y$ that shifts the degree by $\dim Y - \dim X \,\, (\text{mod }2)$. This appears more general then the cases in the first paragraph, but I'm not sure how to connect it with the Thom class picture.

ABS give the construction of the Thom class when the bundle is of even rank, but what about real vector bundles of odd rank? (This is my first question)

For my work, I want to understand how to pushforward along a principal $S^1$-bundle $p \colon E \to X$, giving a map $p_* \colon K^*(E) \to K^{*-1}(X)$ (I actually want this for twisted K-theory, but I'm starting simple). Using the Pontryagin-Thom construction, it is sufficient to construct a Thom class for the normal bundle of $E$ embedded into a larger space. My next question: is there some condition that guarantees that this normal bundle has a $Spin^c$-structure and hence a K-orientation?

Overall, I'm trying to understand pushforwards in K-theory that have a degree shift. Most of the literature appears to discuss the cases in the first paragraph, which deal with even dimensional bundles.

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  • $\begingroup$ Precisely what type of space do you want $E$ embedded into? Obviously if I embed $E$ into itself its normal bundle is 0-dimensional so it is oriented with respect to anything. $\endgroup$ Sep 2, 2021 at 15:22
  • $\begingroup$ To construct the pushforward of $p \colon E \to B$ using the Pontryagin-Thom construction you would factor $p$ as $E \hookrightarrow B \times \mathbb R^n \to B$, using an embedding (e.g. Whitney) of $E$ into $\mathbb R^n$ for some large enough $n$. See ncatlab.org/nlab/show/fiber+integration $\endgroup$
    – Motmot
    Sep 3, 2021 at 6:41
  • $\begingroup$ Maybe ABS, Theorem 12.3 is interesting for you. At the end of §12 they claim, they improved the results of Atiyah and Hirzebruch. $\endgroup$
    – Jonas
    Oct 12, 2021 at 9:27

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The Thom-Isomorphism has a degree shift, if you use the natural $\mathbb{Z}$ grading on $K^*$, instead of the $\mathbb{Z}/2\mathbb{Z}$ grading.

For your next questions, let $f:X \to Y$ be an embedding of oriented smooth manifolds. So we have $TX \overset{\sim}{=} f^*TY \oplus N(f)$, where $N(f) \to X$ is the normal bundle of $f$. Using the multiplicativity of Stiefel-Whitney classes, we can see that $f$ is a $c_1$-map iff $N(f)$ admits a $Spin^c$ structure.

Now additionally assume $Y=\mathbb{R}^n$. So $TX \overset{\sim}{=} N(f)\oplus n$, so $N(f)$ admits a $Spin^c$ structure iff $TX$ does. Thats the best condition I can think of.

Im afraid I have no answer to your first question.

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