Projective spaces

We can also define spaces in other ways, and then try to find cell complex structures for them. For example, the real projective n-space \({\mathbb{R}\textrm{P}^{n}}\) is defined as the space of all lines through the origin in \({\mathbb{R}^{n+1}}\). Each such line is determined by a unit vector, except that the negative of every vector is identified with the same line, so we can consider \({\mathbb{R}\textrm{P}^{n}}\) to be \({S^{n}}\) with antipodal points identified.

Alternatively, we can look at \({\mathbb{R}\textrm{P}^{n}}\) as the unit vectors in the upper hemisphere only, since the lower hemisphere is made up of all negatives of the upper; except that now, antipodal points of the boundary are identified. But the upper hemisphere is \({D^{n}}\) and its boundary is \({S^{n-1}}\) with antipodal points identified, or \({\mathbb{R}\textrm{P}^{n-1}}\). Thus \({\mathbb{R}\textrm{P}^{n}}\) is obtained by attaching an \({n}\)-cell to \({\mathbb{R}\textrm{P}^{n-1}}\), and by induction we can see that \({\mathbb{R}\textrm{P}^{n}}\) has a cell complex structure with one cell in each dimension up to \({n}\).

19.projective-space

The identifications in the above constructions of \({\mathbb{R}\textrm{P}^{2}}\) are not easily visualized, since they cannot be embedded in \({\mathbb{R}^{3}}\); in contrast, \({\mathbb{R}\textrm{P}^{1}}\) is \({D^{1}}\) with boundary \({S^{0}}\) having antipodal points identified, i.e. the line segment with the endpoints identified; in other words \({\mathbb{R}\textrm{P}^{1}=S^{1}}\), the circle.

We can also define the complex projective n-space \({\mathbb{C}\textrm{P}^{n}}\), which is the space of all lines through the origin in \({\mathbb{C}^{n+1}}\). In this case one has a cell complex structure with one cell in each even dimension up to 2\({n}\). \({\mathbb{H}\textrm{P}^{n}}\) can similarly be defined, but \({\mathbb{O}\textrm{P}^{n}}\) can only be defined for \({n<3}\) due to lack of associativity. By generalizing the reasoning above, we have \({\mathbb{C}\textrm{P}^{1}=S^{2}}\); \({\mathbb{H}\textrm{P}^{1}=S^{4}}\); and \({\mathbb{O}\textrm{P}^{1}=S^{8}}\). As manifolds, the projective spaces are all closed, i.e. compact and without boundary.

An Illustrated Handbook