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<p>We show that there does not exist a <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(C_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-ring spectrum whose underlying ring spectrum is <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\textrm{MSpin}^c\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>MSpin</mtext> <mi>c</mi> </msup> </math></EquationSource> </InlineEquation> and whose <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(C_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-fixed point spectrum is <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\textrm{MSpin}\)</EquationSource> <EquationSource Format="MATHML"><math> <mtext>MSpin</mtext> </math></EquationSource> </InlineEquation>. As a corollary, the <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\hbox {spin}^c\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mtext>spin</mtext> <mi>c</mi> </msup> </math></EquationSource> </InlineEquation> and spin orientations of Atiyah–Bott–Shapiro cannot be obtained as the underlying and fixed point maps of a single map of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(C_2\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-ring spectra.</p>

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On the nonexistence of a Green functor with values \(\hbox {spin}^c\) bordism and spin bordism

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月29日

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  • Hassan H. Abdallah,
  • Zachary Halladay,
  • Yigal Kamel

摘要

We show that there does not exist a \(C_2\) C 2 -ring spectrum whose underlying ring spectrum is \(\textrm{MSpin}^c\) MSpin c and whose \(C_2\) C 2 -fixed point spectrum is \(\textrm{MSpin}\) MSpin . As a corollary, the \(\hbox {spin}^c\) spin c and spin orientations of Atiyah–Bott–Shapiro cannot be obtained as the underlying and fixed point maps of a single map of \(C_2\) C 2 -ring spectra.