Completed \(K\)-theory and equivariant elliptic cohomology. (English) Zbl 1505.55016
In this paper the author constructs \(S^1\)-completed \(K\)-theory. The motivation of this theory is Kitchloo and Morava’s idea of Tate \(K\)-theory, which is constructed as the circle-equivariant \(K\)-theory of the free loop space. The author applies this idea and constructs the new theory. Instead of the free loop space, he applies the loop groupoid. In this way the \(S^1\)-completed \(K\)-theory is constructed as a twisted cohomology theory.
The author constructs a new model of equivariant elliptic cohomology at the Tate curve using the theory. In addition, he compares Grojnowski’s equivariant elliptic cohomology with the \(S^1\)-completed \(K\)-theory.
Due to the relation between Tate \(K\)-theory and \(K\)-theory, it is natural to seek a construction of \(S^1\)-completed \(K\)-theory in term of Freed-Hopkins-Telemann’s construction of twisted \(K\)-theory [D. S. Freed et al., J. Topol. 4, No. 4, 737–798 (2011; Zbl 1241.19002), J. Am. Math. Soc. 26, No. 3, 595–644 (2013; Zbl 1273.22015) and Ann. Math. (2) 174, No. 2, 947–1007 (2011; Zbl 1239.19002)]. In this way the author formulates the conclusions about the calculation of the theory as Freed, Hopkins and Telemann did in their papers.
It is worth mentioning that the \(S^1\)-completed \(K\)-theory is the twisted quasi-elliptic cohomology defined by the reviewer and M. B. Young [“Twisted Real quasi-elliptic cohomology”, Preprint, arXiv:2210.07511]. The twisted Real quasi-elliptic cohomology is a Real version of this theory.
The author constructs a new model of equivariant elliptic cohomology at the Tate curve using the theory. In addition, he compares Grojnowski’s equivariant elliptic cohomology with the \(S^1\)-completed \(K\)-theory.
Due to the relation between Tate \(K\)-theory and \(K\)-theory, it is natural to seek a construction of \(S^1\)-completed \(K\)-theory in term of Freed-Hopkins-Telemann’s construction of twisted \(K\)-theory [D. S. Freed et al., J. Topol. 4, No. 4, 737–798 (2011; Zbl 1241.19002), J. Am. Math. Soc. 26, No. 3, 595–644 (2013; Zbl 1273.22015) and Ann. Math. (2) 174, No. 2, 947–1007 (2011; Zbl 1239.19002)]. In this way the author formulates the conclusions about the calculation of the theory as Freed, Hopkins and Telemann did in their papers.
It is worth mentioning that the \(S^1\)-completed \(K\)-theory is the twisted quasi-elliptic cohomology defined by the reviewer and M. B. Young [“Twisted Real quasi-elliptic cohomology”, Preprint, arXiv:2210.07511]. The twisted Real quasi-elliptic cohomology is a Real version of this theory.
Reviewer: Zhen Huan (Wuhan)
MSC:
| 55N34 | Elliptic cohomology |
References:
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