Subject: Response to Claude Schochet
From: Mark Hovey <mhovey@wesleyan.edu>
Date: 16 Aug 2005 11:18:18 -0400

Claude,
  In the paper "Spin cobordism determines real K-theory" by Mike Hopkins
and myself, we prove that

MSpin^c_*(X) tensor_{MSpin^c_*} KO_* \cong KO_* X

just as you want.  This was one of the first papers I wrote; in my
opinion, the description of this proof contained in "Spin bordism and
elliptic homology" is much clearer.  Here are the two BibTeX references.
              Mark

@article {MR1166518,
    AUTHOR = {Hopkins, Michael J. and Hovey, Mark A.},
     TITLE = {Spin cobordism determines real {$K$}-theory},
   JOURNAL = {Math. Z.},
  FJOURNAL = {Mathematische Zeitschrift},
    VOLUME = {210},
      YEAR = {1992},
    NUMBER = {2},
     PAGES = {181--196},
      ISSN = {0025-5874},
     CODEN = {MAZEAX},
   MRCLASS = {55N22 (19L41)},
  MRNUMBER = {MR1166518 (93g:55007)},
MRREVIEWER = {P. S. Landweber},
}

@article {MR1337212,
    AUTHOR = {Hovey, Mark A.},
     TITLE = {Spin bordism and elliptic homology},
   JOURNAL = {Math. Z.},
  FJOURNAL = {Mathematische Zeitschrift},
    VOLUME = {219},
      YEAR = {1995},
    NUMBER = {2},
     PAGES = {163--170},
      ISSN = {0025-5874},
     CODEN = {MAZEAX},
   MRCLASS = {55N22 (55N20 57R15)},
  MRNUMBER = {MR1337212 (96j:55006)},
MRREVIEWER = {P. S. Landweber},
}