On the nonexistence of a Green functor with values MSpin${}^c$ and MSpin

The paper demonstrates a significant non-existence result in equivariant stable homotopy theory: there does not exist a genuine \(C_2\)-ring spectrum \(E\) whose underlying spectrum is the MSpin spectrum, and whose \(C_2\)-fixed point spectrum is also the MSpin spectrum. This result is stronger than previous results because it drops an assumption on the preservation of the Â-genus under the restriction map.

The key innovation is a new proof based solely on the algebraic structure of the homotopy groups of MSpin and MSpin, as elucidated in previous work by one of the authors. The argument involves showing that if such a \(C_2\)-spectrum existed, certain elements in the homotopy groups of MSpin would have to satisfy conditions dictated by the Mackey functor structure. By carefully analyzing the ring structure of MSpin modulo 2 and using the Anderson-Brown-Peterson theorem, the authors show that these conditions lead to a contradiction, implying the non-existence of the desired \(C_2\)-spectrum.

Prior ingredients include:
1. The construction of a genuine \(C_2\)-commutative ring spectrum MSpin\(\mathbb{R}\) in previous work by Halladay and Kamel, whose underlying spectrum is MSpin and whose \(C_2\)-fixed points receive a map from MSpin. This paper highlighted the fact that the natural map from MSpin to the \(C_2\)-fixed points of MSpin\(\mathbb{R}\) is not an equivalence.
2. A description of the algebraic structure of the abelian groups MSpin and the ring MSpin given in Abdallah and Salch’s work.
3. The Anderson-Brown-Peterson theorem describing the structure of the Spin cobordism ring.