Fundamental group

A fundamental group is a concept in algebraic topology. A fundamental group is a group of equivalence classes (a set with things that are the same) of loops in an area. This means that the group gives information about loops in an area.  The fundamental group is the simplest type of homotopy group. The fundamental group is also an "homotopy invariant". The fundamental group of a topological space  is written as an .

 Intuition 
Think of a space. For example, one could start with a surface. In that space, there is a point in that space. All loops in this space start and end at this point. A line can start at this point. The line will then move around and goes back to the starting point. Also, two different loops can be combined. Two loops are thought to be the same if they can be combined into each other without breaking. The set all loops that can be combined and be equal is called the fundamental group.

References











 Peter Hilton and Shaun Wylie, Homology Theory, Cambridge University Press (1967) [warning: these authors use contrahomology for cohomology]





 Deane Montgomery and Leo Zippin, Topological Transformation Groups, Interscience Publishers (1955)








Other websites

 Dylan G.L. Allegretti, Simplicial Sets and van Kampen's Theorem: A discussion of the fundamental groupoid of a topological space and the fundamental groupoid of a simplicial set
 Animations to introduce fundamental group by Nicolas Delanoue
 Sets of base points and fundamental groupoids: mathoverflow discussion
 Groupoids in Mathematics

Category:Topology