In work related to positive scalar curvature, Stephan Stolz (Ann. Math.), but later Stolz-Kreck ($HP2$ bundles and Elliptic cohomology) introduced a version of Real connective $K$-homology by considering spin cobordism, localizing at the prime $2$ and then killing classes which are determined by bundles with fiber $HP2$.
Is there an analogous version for complex (connective) $K$-homology?
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3$\begingroup$ Maybe you are interested in Hopkins, Michael J.(1-MIT); Hovey, Mark A.(1-YALE) Spin cobordism determines real K-theory. Math. Z. 210 (1992), no. 2, 181–196. which proves there is an iso $MSpin^c_*(X)\otimes_{MSpin^c_*}K_*\to K_*(X)$ ? $\endgroup$– nsrtCommented Feb 7, 2014 at 15:15
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$\begingroup$ And, as Hopkins-Hovey discuss, for the complex orientation $MU \to KU$ this goes back to Connor-Floyd in 1966, see here: ncatlab.org/nlab/show/… $\endgroup$– Urs SchreiberCommented Apr 24, 2014 at 8:50
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