A new approach to twisted \(K\)-theory of compact Lie groups (Q1985986)

The compact connected Lie groups provide a wide class of well studed objects from the point of view of cohomology theories, but not yet of \(K\)-theories and is a starting point for many theories like homotopy, \(C^\ast\)-algebras, \(K\)-homologies, etc. The twisted \(K\)-theory of compact Lie groups is known in many cases, but not yet all. The paper is devoted to computation problem of twisted \(K\)-theories of compact Lie groups., The author makes a survey of known results and also proposed some new computation by using the Segal spectral sequences. Let \(\mathcal H\) be an infinite dimensional separable Hilbert space, \(\mathbb T \) be a maximal torus in the unitary group \(U(\mathcal H)\). Given a compact topological space \(X\), the principal bundle \(P: PU(\mathcal H) = U(\mathcal H)/\mathbb T \cong \mathbb {CP}^\infty \hookrightarrow E \twoheadrightarrow X\) over \(X\) classified the twisted \(K\)-theories \(K^*(X,h)\) and \(K_*(X,h)\) by the elements \(h\in H^3(X,\mathbb Z)\). Let \(G\) be a connected simple compact Lie group. The 3rd cohomology group \(H^3(G,\mathbb Z) \cong \mathbb Z\) is used as the group of twists. The following results are previously known: \begin{itemize} \item[1.] For simple connected and simply connected compact Lie group \(G\) of rank \(\mathrm{rank}\; G = n\), the twisted \(K\)-homology \(K_\bullet(X,h)\) is a product of an algebra over \(\mathbb Z\) on \(n-1\) odd-degree generators with a finite cyclic group of order \(c(G,h)\) a divisor of \(h>0\) (Theorem 1). \item[2.] In all the cases of simple connected simply connecte compact Lie group of type \(A_n\), \(B_n\), \dots \(E_8\) the order \(c(G,h) =h/\mathrm{gcd}(h,y(G))\) of the torsion of \(K_\bullet(X,h)\) and \(K^\bullet(X,h)\) and \(y(G)\) are conjecturally computed (Theorem 2). \item[3.] There exists a homological Segal spectral sequence \(H_p(B,K_q(F,\imath^*h)) \Longrightarrow K_\bullet(E,h)\) associated to any fiber bundle of CW complexes \( F \xrightarrow{\imath} B \xrightarrow{pr} B \) and a twist \(h\in H^3(G,\mathbb Z)\) (Theorem 3). \end{itemize} The author used this Segal spectral sequences to provide a new method for computing (co)homological twisted \(K\)-theories of rank 2 connected and simply connected compact simple Lie groups. The new results are: \begin{itemize} \item[1.] For the bundle \(F \subset E \twoheadrightarrow B\), the Hurewicz map \(\pi_r(F) \to H_r(B,K_0(F,\imath^*h))\), the high differentials \(d^2,d^3,\dots,d^{r-1}\) leave the 2nd term \(E^2_{r,0}=H_r(B,K_0(F,\imath^*h))\) unchanged (Theroem 6) \item[2.] The Hurewicz map \(\pi_4(\mathbb S^3) \to K_4(\mathbb S^3,h) \cong \mathbb Z/h\), \(h\ne 0\), is nonzero if and only if \(h\in \mathbb Z\) is even (Theorem 9). \item[3.] The Hurewicz maps \(\pi_j(\mathbb S^3) \cong \pi_j(P_h) \to ku_j(P_h)\) are injective on several cases (Theorem 16). \item[4.] For an odd or \(2 \mod 4\) twist \(h\) the odd torsion of the \(K\)-homology \(K_\bullet(PSp(2),h)\) of the projective symplectic group \(PSp(2)\) and twist \(h\) is a cyclic group of order \(h_{odd}/gcd(h,3)\), and is \((\mathbb Z/2)^4\) is \(h\) is \(2 \mod 4\) (Theorem 17). \end{itemize} The paper is technical but gives a good survey of the problem.