Vipul at 23:18, 1 April 2011

2011-04-01T23:18:42Z

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	The fundamental group of a [[~~smash~~ product]] is the product ~~(in the usual sense)~~ of ~~the~~ fundamental groups~~. It also turns out that the fundamental group of the usual product of topological spaces (which is not the correct notion of product for based topological spaces) is the same~~.		The fundamental group of a [[product of based topological spaces]] is the product of their fundamental groups.

	{{coproduct-preserving functor}}		{{coproduct-preserving functor}}

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==Definition==

The '''fundamental group functor''' <math>\pi_1</math> is a [[functor]] from the [[category of based topological spaces with based maps]] to the [[category of groups]]. The functor is defined as follows:

* It sends each topological space <math>X</math> to its [[fundamental group]] <math>\pi_1(X)</math>
* Given a map <math>f:X \to Y</math> of topological spaces, the associated morphism of fundamental groups is defined as follows: <math>\pi_1(f)(\gamma) = f \circ \gamma</math> upto homotopy.

If <math>f,g: X \to Y</math> are homotopic maps, then they induce the same homomorphism on fundamental groups. Thus, the fundamental group functor descends to a functor from the [[homotopy category of based topological spaces]] to the category of groups.

==Properties of the functor==

{{product-preserving functor}}

The fundamental group of a [[smash product]] is the product (in the usual sense) of the fundamental groups. It also turns out that the fundamental group of the usual product of topological spaces (which is not the correct notion of product for based topological spaces) is the same.

{{coproduct-preserving functor}}

The fundamental group of a [[wedge sum]] is the direct sum of the fundamental groups.

{{non-faithful functor}}

The fundamental group functor is far from faithful. Firstly, two different maps which are homotopic, induce the same homomorphism on fundamental groups. Even if we go down to the homotopy category, there could be many non-homotopic maps that induce the same map on fundamental groups.

{{non-full functor}}

The fundamental group functor is not full either. In other words, given two topological spaces, every homomorphism between their fundamental groups need not be realized as arising from a continuous map between the topological spaces.

===Surjectivity on objects===

Every [[group]] can be realized as the fundamental group of a [[topological space]]. More strongly, associated with every group, there is a [[classifying space]], viz a space whose universal covering space is [[contractible space|contractible]], and whose fundamental group is the given group.

Fundamental group functor - Revision history

Vipul at 23:18, 1 April 2011

Vipul: 1 revision

Vipul at 23:07, 2 November 2007