Comparing orthogonal calculus and calculus with reality (Q6585670)

Let \(F\) be a functor from the category of Euclidean spaces to (based) spaces. Orthogonal calculus assigns to \(F\) a tower of functors \N\[\N\cdots \rightarrow T_nF\rightarrow T_{n-1}F\rightarrow \cdots \rightarrow T_1F\rightarrow T_0F. \N\]\NThe \(n\)-th layer of this tower is the homotopy fibre \(D_nF\) of the map \(T_nF\to T_{n-1}F\). Motivated by the analogy between orthogonal calculus and real topological \(K\)-theory, the author developed unitary calculus [\textit{N. Taggart}, J. Homotopy Relat. Struct. 17, No. 3, 419--462 (2022; Zbl 1514.55009)] and calculus with Reality [\textit{N. Taggart}, Glasg. Math. J. 64, No. 1, 197--230 (2022; Zbl 1480.55013)]. Unitary calculus is analogous to complex topological \(K\)-theory, while calculus with Reality is the calculus version of Atiyah's \(K\)-theory with Reality.\N\NThe three calculi are related the same way as their analogous \(K\)-theories. The complexification-realification adjunction on the level of vector spaces induces a relationship between orthogonal and unitary calculus. Forgetting the complex conjugation action on calculus with Reality recovers unitary calculus. In this paper, the author considers the relationship between calculus with Reality and orthogonal calculus. He proves the calculus version of the fact that the spectrum of real topological \(K\)-theory is the (homotopy) \(C_2\)-fixed points of Atiyah's \(K\)-theory with Reality spectrum.\N\NThe major technical difficulty to overcome is the non-existence of a good fixed points functor. Calculus with Reality is constructed with respect to the underlying non-equivariant equivalences of \(C_2\)-spaces, which are preserved by homotopy \(C_2\)-fixed points, but in general not by \(C_2\)-fixed points. Thus, the author models \(C_2\)-spaces by cofree \(C_2\)-spaces, on which homotopy \(C_2\)-fixed points and \(C_2\)-fixed points agree. He then applies this to the equivalence between free and cofree \(C_2\)-spaces to produce a cofree model for calculus with Reality.