The aim of this note is to compute the fundamental group of using quaternions. More precisely, an isomorphism will be found leading to:
Property 1: is the univeral covering of and .
Before introducing quaternions, recall the construction of . Usually, is defined as the splitting field of the polynomial over , namely . Taking the companion matrix of , we get an isomorphism with a subalgebra of . Thus, a possible construction of the quaternions is to replace real coefficients with complex ones, namely . Notice that if , , and , then every element of can be uniquely written as a -linear combination of , , and .
Formally, is a four-dimensional real algebra with a basis satisfying ; in particular, and , so is not commutative. In fact, it can be shown that is a noncommutative division algebra.
Viewing as a subalgebra of , there are a natural conjugaison and a natural norm corresponding respectively to the conjugaison and the determinant in . As a real vector space, the norm define a definite positive quadratic form on , whose bilinear form is . In particular, notice that and are orthogonal.
Now let act on by conjugaison, ( is the group of automorphisms of viewed as a real vector space). Since and each preserve the quadratic form , taking the restrictions of the ‘s to we can suppose that . Moreover, is connected and is continuous so .
To show that is onto, notice that is and that induces a linear map between the Lie algebra and ; by connectedness of and using the inverse function theorem, it is sufficient to show that is invertible. But and . Therefore, is onto.
Noticing finally that , we get the isomorphism . But the isomorphism is also a homeomorphis since the coordinate functions of are just rational functions, and identifying with the element of of norm , is homeomorphic to the sphere ; in particulier, is simply connected. Moreover, the action of on by left multiplication is properly discontinuously, so we can deduce property 1.
In fact, the isomorphism has a topological interpretation:
Property 2: is homeomorphic to the projective space .
By definition, is the quotient of by the equivalence relation . In fact, all points of are not needed, and is also homeomorphic to the quotient of by ; since the projection induces a bijection from to , we get that is also homeomorphic to quotiented by the symmetry with respect to .
Now let where denotes the rotation of axis and angle . Then is a continuous surjection such that for all , is equivalent to . Therefore, induces a continuous bijection ; since is compact we deduce that is in fact a homeomorphism.