Overview and basic definition
Definition: Real projective space
Define an equivalence relation on by setting if and only if are on the same line through the origin; precisely, this means that there exists such that . Then and are sometimes said to be antipodal.
The -dimensional real projective space is the “space of lines in .”
In point-set topology
- Homeomorphisms and topological embeddings: The real projective space is homeomorphic to the following quotient spaces:
- Topological manifolds: The homeomorphisms above imply that is a topological -manifold, i.e., locally Euclidean, Hausdorff, and second-countable (has a countable basis).
In algebraic topology
- Cell attachments and complexes: has the cell complex structure with one cell in each dimension .
- Fundamental group of the circle: Since , we know that it has fundamental group .
- Calculating fundamental groups: This theorem can be applied to calculate the fundamental group when by “guessing” a homeomorphism , where is the cyclic group of order 2 whose generator acts on by . Thus,
Notes
Homeomorphism with
Claim: The real projective space is locally -Euclidean.
Proof from Topology.
For , define the set of equivalence classes
This is well-defined since if , then for all . Each is open since
is open in (see the following commutative diagram). We will show that for all , then the claim follows from the fact that .
concept-question Why is the vector -dimensional?
wip Define by for all . Note that since both numerator and denominator will be scaled by , so induces a map satisfying
The map is bijective since we can define an explicit inverse
As a concrete example, for we have
.
As a topological manifold
Theorem ( Topology HW 8.4): The real projective space is homeomorphic to quotients of spheres and closed disks
The real projective space is homeomorphic to the following quotient spaces:
- (i) , the unit sphere in with antipodal points identified;
- (ii) , where is the closed unit disk centered at the origin and the equivalence classes are given by if .
Thus, is a topological n-manifold.
Facts:
- Sn quotient map is open
- Sn Hausdorff.