Mathematics Prelims

March 18, 2009

The Fundamental Group, Part II – The Basepoint

Filed under: Topology — cjohnson @ 2:07 pm

We constructed the fundamental group by looking at the homotopy classes of loops with a fixed basepoint.  The question that naturally arises is how does the fundamental group change if we move the basepoint.  To begin let’s suppose we’re in a path connected space X and we have two basepoints we’re concerned about, say x_0 and x_1.  Since X is path connected, there exists some path f from x_0 to x_1.  We can use this to convert x_1 loops into x_0 loops.

What we’ll do is take a loop, say g, based at x_1 and use our path to get from x_0 to x_1, follow the loop, then take the reverse path back to x_0.  This is illustrated in the image below (adapted from an image in Hatcher’s book).

Connected Loops

This new path f * g * \overline{f} is a loop based at x_0.  Now, if two x_1-loops, say g_1 and g_2 are homotopic, then so are the associated x_0-loops: g_1 \simeq g_2 \implies f * g_1 * \overline{f} \simeq f * g_2 * \overline{f}.

If we now construct a map \beta_f : \pi_1(X, x_1) \to \pi_1(X, x_0) by setting \beta_f [g] = [f * g * \overline{f}] we see that the map will be well-defined since our construction of the x_0-loops respects homotopy.  But what happens when we concatenate two x_1-loops?

Suppose g and h are loops based at x_1.  We want to see what this map does to the homotopy class of g * h.

\displaystyle \beta_f([g] \cdot [h]) = \beta_f[g * h]

\displaystyle = [f * g * h * \overline{f} ]

\displaystyle = [f * g * \overline{f} * f * h * \overline{f} ]

\displaystyle = [f * g * \overline{f}] \cdot [f * h * \overline{f}]

\displaystyle = \beta_f [g] \cdot \beta_f [h]

So our \beta_f map is in fact a homomorphism.  Notice that f * g * \overline{f} \simeq f * h * \overline{f} implies g \simeq h, which means that our map is injective.

To show the map is surjective, let h be an x_0-loop, then \overline{f} * h * f is an x_1-loop.  Applying \beta_f we have \beta_f[\overline{f} * h * f] = [f * \overline{f} * h * f * \overline{f}] = [h], and so our map is surjective.  Together with injectivity above we have shown that in a path connected space the fundamental group of \pi_1(X, x_0) and \pi_1(X, x_1) are isomorphic.  We then simply write \pi_1(X) since the choice of basepoint is basically irrelevant.

If our space is not path connected, but x_0 and x_1 are in the same path component, the above gives us that \pi_1(X, x_0) \cong \pi_1(X, x_1).


1 Comment »

  1. […] The Fundamental Group, Part 2 – The Basepoint […]

    Pingback by Topology, Geometry & Dynamics | Topology, Geometry & Dynamics — September 24, 2010 @ 4:57 pm | Reply

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