Show that \(SP_2(S^1)\) is a Möbius band, and that this is consistent with the description of \(SP_2(S^n)\) as a mapping cone given in Example 4K.5.
Solution
Represent an unordered pair of points of \(S^1\subset\mathbb C\) by the chord joining them. Thus \[ \{z,w\}\longmapsto [z,w] \] identifies \(SP_2(S^1)\) with the space of unoriented chords of the circle, including degenerate chords \(z=w\).
A nondegenerate chord is determined by two parameters: the unoriented line through the origin perpendicular to the chord, and the signed distance of the chord from the origin along this line. Equivalently, choose an angle \(\theta\) for a unit normal and a number \(s\in[-1,1]\). Reversing the normal changes \((\theta,s)\) to \[ (\theta+\pi,-s). \] Hence the chord space is \[ (S^1\times[-1,1])/\bigl((\theta,s)\sim(\theta+\pi,-s)\bigr), \] which is the standard interval-bundle model of the Möbius band. The boundary \(|s|=1\) consists exactly of the degenerate chords \(\{z,z\}\), so the diagonal copy of \(S^1\) is the boundary circle of the Möbius band. Therefore \[ \boxed{SP_2(S^1)\cong\text{Möbius band}.} \]
This agrees with Example 4K.5. For \(n=1\), the space denoted there by \(S^n\mathbb{RP}^{\,n-1}\) is the unreduced suspension of \(\mathbb{RP}^0\), hence an interval. The example describes \(SP_2(S^1)\) as the mapping cone obtained by attaching the cone on this interval to the diagonal \(S^1\) via the endpoint identification coming from interchanging the two points. Attaching this twisted strip to the diagonal circle is precisely the usual mapping-cone construction of a Möbius band. Thus the two descriptions coincide.