Example of a Topological Space with a Nonabelian Fundamental Group and a Nontrivial Center
In the realm of topology, the fundamental group is a significant concept used to classify topological spaces. This article delves into an example of a topological space that has both a nonabelian fundamental group and a nontrivial center. We will explore the construction of such a space and a specific algebraic group that fits this description. By understanding these concepts, we aim to provide a deeper insight into advanced topics in algebraic topology.
Introduction to Topological Spaces and Fundamental Groups
A topological space is a mathematical structure that generalizes the notion of a geometric space. The fundamental group, denoted as (pi_1), of a topological space is an algebraic invariant that captures information about the loops in the space and their homotopy classes. When the space is a CW complex, which is a type of topological space built by attaching cells of varying dimensions, the fundamental group can often be explicitly described.
Constructing a Topological Space with a Desired Fundamental Group
Given any group (G), one can construct a topological space (X) such that its fundamental group is isomorphic to (G). This can be achieved by forming a CW complex with a certain number of generators and relations. The example below illustrates this construction using a nonabelian group (G).
Example: Constructing a Topological Space with (pi_1X cong G)
Consider a group (G langle g_{alpha}, r_{beta} rangle) with a presentation derived from the Universal Property of Free Groups. This presentation allows us to form a topological space (X) from a wedge sum of circles (X bigvee_{alpha} S^1_{alpha}). Two-cells are attached to (X) along the relations (r_{beta}), creating a complex where the Cayley graph of (G) serves as the universal cover. The orbit space of this cover under the action of (G) is exactly (X).
Presentation and Construction
The group (G) can be described by a presentation (G langle g_{alpha}, r_{beta} rangle). The space (X) is constructed from a wedge sum of circles, and two-cells are attached along the relations (r_{beta}). The resulting space (X) has a fundamental group (pi_1X cong G). This construction is based on the fact that the universal cover of (X), denoted (tilde{X}), is simply connected, and the fundamental group of (X) is isomorphic to (G).
The Example: SL2C/SL2Z
A concrete example of a topological space with a nonabelian fundamental group and a nontrivial center is the quotient space SL2C/SL2Z. Let us explore this construction in detail.
Defining the Groups
The group SL2C consists of 2x2 complex matrices with determinant 1. It is a topological group inherited from the topological vector space C4. The group SL2Z is the set of 2x2 integer matrices with determinant 1. This is a subgroup of SL2C.
Group Action and Quotient Space
A group action is defined on SL2C by multiplying elements of SL2Z. The equivalence classes are then formed by identifying two elements (a) and (b) if there exists a (z) in SL2Z such that (b za). This defines the quotient space SL2C/SL2Z.
Properties of the Quotient Space
For the fundamental group of SL2C/SL2Z to be isomorphic to SL2Z, the group action of SL2Z on SL2C must be free and proper. The free action implies that no non-identity element of SL2Z fixes any point in SL2C. The proper action means that for each (a) in SL2C, there exists a neighborhood such that for any non-identity (g) in SL2Z, the image of the neighborhood under (g) does not intersect the original neighborhood.
Conclusion on Fundamental Group
Given that SL2C is contractible to the 3-sphere SU2, its fundamental group is trivial. The action of SL2Z on SL2C is free and proper, as SL2Z is a discrete subgroup. Therefore, the fundamental group of the quotient space SL2C/SL2Z is isomorphic to SL2Z. Additionally, SL2Z is a nonabelian group with a nontrivial center, being the subgroup {I, -I}.
Conclusion
This article has explored the construction and properties of a topological space with a nonabelian fundamental group and a nontrivial center. By understanding the construction of the fundamental group and the properties of specific examples like SL2C/SL2Z, we can gain deeper insights into advanced topics in algebraic topology.
References
[1] Hatcher, A. (2002). Algebraic Topology. Cambridge University Press.