- #1
seydunas
- 39
- 0
Hi
i want to see why the fundamental group of lens space L(p,q) is Z_p. Can you help me?
i want to see why the fundamental group of lens space L(p,q) is Z_p. Can you help me?
seydunas said:Hi
i want to see why the fundamental group of lens space L(p,q) is Z_p. Can you help me?
The fundamental group of a lens space is a mathematical concept used in topology to describe the fundamental building blocks of the space. It is a group of all the possible loops that can be made in the space, where the starting and ending point of the loop are fixed. In other words, it is the set of all possible paths that can be taken in the space, up to homotopy equivalence.
The fundamental group of a lens space can be calculated using the Seifert-van Kampen theorem, which states that the fundamental group of a space can be obtained by combining the fundamental groups of smaller subspaces. In the case of a lens space, it can be calculated by decomposing the space into simpler spaces, such as circles, and then combining their fundamental groups using certain operations.
The fundamental group of a lens space is an important topological invariant, meaning that it does not change even if the space is stretched, twisted, or deformed. It helps in distinguishing one lens space from another and is often used in classifying spaces in topology. It also has applications in other areas of mathematics, such as algebraic geometry and differential geometry.
Yes, the fundamental group of a lens space can be infinite. This depends on the specific lens space in question. For example, the fundamental group of the 3-dimensional lens space L(3,1) is infinite, while the fundamental group of the 3-dimensional lens space L(3,2) is finite.
The fundamental group of a lens space is the first homotopy group and is related to the higher homotopy groups. It is also related to the first homology and cohomology groups of the space. In particular, the first homology group is isomorphic to the abelianization of the fundamental group, while the first cohomology group is dual to the fundamental group.