What is the fundamental group of the projective plane?
Z/2Z
The projective plane RP2 has fundamental group Z/2Z since it is the quotient of S2 by making the identifications x = -x. The projection map is a covering map, and the group of covering transformations is just Z/2Z = {id, x -> -x}.
What is fundamental group of covering space?
The fundamental group is one of the most important topological invariants of a space, and a rather accessible one at that. It is essentially a “group of loops,” consisting of all possible loops in a space up to homotopy. Definition 2.1. A loop (sometimes called a closed path) in X is a path f with f(0) = f(1).
What is a Galois covering?
We say that a covering f : Y → X is a Galois covering, if Y is connected and G = Aut(f) acts transitively on f−1(x) for some (and thus any) x ∈ X.
What is the fundamental group of a topological space?
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space.
Is projective space a group?
the quotient group of GL(V) modulo the matrices that are scalar multiples of the identity. (These matrices form the center of Aut(V).) The groups PGL are called projective linear groups.
Is the homology of real projective space the same?
The proof follows from fact (1). 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.
Which is the fundamental group of the projective n-space?
The projective n -space is compact, connected, and has a fundamental group isomorphic to the cyclic group of order 2: its universal covering space is given by the antipody quotient map from the n -sphere, a simply connected space. It is a double cover.
Which is a special case of the real projective space?
In mathematics, real projective space, or RP n or P n ( R ) {displaystyle mathbb {P} _{n}(mathbb {R} )} , is the topological space of lines passing through the origin 0 in R n+1. It is a compact, smooth manifold of dimension n, and is a special case Gr(1, R n+1) of a Grassmannian space.
Which is the first homology group of a space?
: The space is homeomorphic to the circle , and the fundamental group is isomorphic to . This is also the first homology group. : The fundamental group is , because its double cover is simply connected (see n-sphere is simply connected for n greater than 1 ). The first homology group is also .