Free involutions on \(S ^{1} \times S ^{n }\)
The authors study the classification problem of free involutions on \(S^1\times S^n\) up to conjugation for \(n\geq 3\). It is known that the quotients of the four simplest standard involutions are \(S^1\times S^n\), \(S^1\tilde{\times} S^n\), \(S^1\times \mathbb R\text{P}^n\) or the connected sum \(\mathbb R\text{P}^{n+1}\#\mathbb R\text{P}^{n+1}\), where \(S^1\tilde{\times} S^n\) denotes the non-trivial \(S^n\)-bundle over \(S^1\). First, the authors show that any free involution of \(S^1\times S^n\) is homotopically conjugate to one of the four standard involutions. Next, they prove that two free involutions on this space are topologically conjugate if and only if the two quotients are homeomorphic. In particular, they show that every involution is topologically conjugate to the standard involution if its quotient is of type \(S^1\times S^n\) or of type \(S^1\tilde{\times} S^n\). Moreover, they prove that there is a one-one correspondence between the set of conjugacy classes of free involutions and \[ \begin{cases} (\mathbb Z /2)^{\lfloor \frac{n-1}{2}\rfloor +\lfloor \frac{n-2}{4}\rfloor} &\text{ if }n\not\equiv 3 \mod 4\\ \mathbb Z_{\geq 0}\times (\mathbb Z/2)^{\frac{3n-2}{4}} &\text{ if }n\equiv 3 \mod 4 \end{cases} \] if its quotient is of type \(S^1\times \mathbb R\text{P}^n\). Their proof is based on surgery theory and some computations of homotopy groups by using the Fedeler-Schults spectral sequence.
scientific article; zbMATH DE number 5963442
scientific article