A topological characterization of \(SU\) (Q1279750)

Let \(X\) be a CW-complex of finite type which is a homotopy associative \(H\)-space, and which has the following three properties: (1) \(X\) is simply connected, (2) \(H^*(X;\mathbb{Z})\) is an exterior algebra generated by elements \(x_{2i+1}\) of dimensions \(2i+1\), \(i=1,2, \dots\), (3) there exist maps \(j':\Sigma C\mathbb{P}^\infty\to X\) and \(\lambda': \Sigma^2X\to X\) such that the induced homomorphism \(j'{}^*\) of the cohomology is epic and \((\lambda' \circ (\Sigma^2j'))^* (x_{2i+i}) =\pm is_3t^{i-1}\), where \(s_3\) is a generator of \(H^*(S^3; \mathbb{Z})\) and \(t\) is a generator of \(H^* (C\mathbb{P}^\infty, \mathbb{Z})\). The author proves (the main theorem) that under these conditions, \(X\) is homotopy equivalent to the double loop space \(\Omega^2 (X\langle 3\rangle)\) of the 3-connected fibre space \(X\langle 3\rangle\) of \(X\). In the case of the special unitary group SU of infinite dimension, the author constructs two maps \(j_{SU}'\) and \(\lambda_{SU}'\) satisfying condition (3), which shows that the result of the main theorem applies to SU. At last the author gives a topological characterization of SU by showing that any space \(X\) satisfying the conditions in the main theorem is homotopy equivalent to SU.