What is the dimension of a projective space?
So, a projective space of dimension n can be defined as the set of vector lines (vector subspaces of dimension one) in a vector space of dimension n + 1. A projective space can also be defined as the elements of any set that is in natural correspondence with this set of vector lines.
Is the real projective plane compact?
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself.
Is real projective space a manifold?
Real projective spaces are smooth manifolds.
What is the cohomology of real projective space?
Cohomology of real projective space. This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is cohomology group and the topological space/family is real projective space.
Is the homology of projective space the same as the chain complex?
By fact (1), we note that the homology of real projective space is the same as the homology of the following chain complex, obtained as its cellular chain complex: where the largest nonzero chain group is the chain group. Note that the multiplication maps alternate between multiplication by two and multiplication by zero.
Which is the invariant of real projective space?
The invariant is cohomology group and the topological space/family is real projective space Here, denotes the 2-torsion subgroup, i.e., the subgroup comprising elements of order dividing 2.
How does the cohomology of an elementary abelian group work?
Suppose is an elementary abelian 2-group, i.e., a group in which the double of every element is zero. Then, (so ) and , and we get: This in particular applies to the case that is the group , i.e., when we are taking coefficients in the field of two elements. We illustrate how the cohomology groups work for small values of .
