Why is the Fundamental Group of Lens Space L(p,q) Z_p?

Click For Summary
SUMMARY

The fundamental group of lens space L(p,q) is definitively Z_p, as established through the covering relationship with the 3-sphere S^3 and the action of the cyclic group Z/pZ. This relationship indicates that lens spaces can be viewed as quotient spaces formed by the action of a cyclic group on S^3, leading to the conclusion that their fundamental group inherits the properties of Z_p. The discussion emphasizes the importance of understanding the topology of lens spaces and their relationship with covering spaces.

PREREQUISITES
  • Understanding of fundamental groups in algebraic topology
  • Familiarity with lens spaces and their properties
  • Knowledge of covering spaces and quotient spaces
  • Basic concepts of cyclic groups and their actions
NEXT STEPS
  • Study the properties of lens spaces L(p,q) in detail
  • Explore the concept of covering spaces in algebraic topology
  • Learn about the relationship between cyclic groups and topological spaces
  • Investigate the implications of fundamental groups in different topological contexts
USEFUL FOR

Mathematicians, particularly those specializing in algebraic topology, students studying topology concepts, and researchers interested in the properties of lens spaces and their fundamental groups.

seydunas
Messages
39
Reaction score
0
Hi

i want to see why the fundamental group of lens space L(p,q) is Z_p. Can you help me?
 
Physics news on Phys.org
seydunas said:
Hi

i want to see why the fundamental group of lens space L(p,q) is Z_p. Can you help me?

I think that lens spaces are covered by the 3 sphere by the action of the cyclic group, Z/pZ
 
So;

i have to ask again why that lens spaces are covered by the 3 sphere by the action of the cyclic group Z/pZ
 

Similar threads

Graduate Why are sheafs defined using abelian groups?
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Is H a Lie Group with Subspace Topology from T^2?
  • · Replies 17 ·
Replies
17
Views
5K
Graduate Fundamental Group of (X,p): D^2\{(x,0) : 0<=x<=1}
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Fundamental definition of extrinsic curvature
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Weird Semidirect Product Formula
  • · Replies 3 ·
Replies
3
Views
1K
Insights Groups, The Path from a Simple Concept to Mysterious Results
  • · Replies 17 ·
Replies
17
Views
9K
Graduate Hi,i want to understand how fundamental group of a closed oriented
  • · Replies 8 ·
Replies
8
Views
3K
Graduate Fundamental Group of the Torus-Figure 8
  • · Replies 3 ·
Replies
3
Views
5K
Graduate How Do You Determine the Fundamental Groups of Specific Matrix Spaces?
  • · Replies 3 ·
Replies
3
Views
4K
Graduate What Does a Trivial Fundamental Group Indicate About Space X?
  • · Replies 1 ·
Replies
1
Views
3K