Algebraic Topology - Fundamental Group and the Homomorphism induced by h

[Math Amateur]

On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachement giving Munkres pages 333-334)

"Suppose that $h: X \rightarrow Y$ is a continuous map that carries the point $x_0$ of X to the point $y_0$ of Y.

We denote this fact by writing:

$h: ( X, x_0) \rightarrow (Y, y_0)$

If f is a loop in X based at $x_0$ , then the composite $h \circ f : I \rightarrow Y$ is a loop in Y based at $y_0$"

I am confused as to how this works ... can someone help with the formal mechanics of this.

To illustrate my confusion, consider the following ( see my diagram and text in atttachment "Diagram ..." )


Consider a point $i^'$ $\in [0, 1]$ that is mapped by f into $x^'$ i.e. $f( i^{'} )$ $= x^'$

Then we would imagine that $i^'$ is mapped by $h \circ f$ into some corresponding point $y^'$ ( see my diagram and text in atttachment "Diagram ..." )


i.e. $h \circ f (i^{'} )$ $= y^'$

BUT

$h \circ f (i^{'} ) = h(f(i^{'} )) = h(x^{'} )$

But (see above) we only know of h that it maps $x_0$ into $y_0$? {seems to me that is not all we need to know about h?}

Can anyone please clarify this situation - preferably formally and explicitly?

Peter

 

[lavinia]

A loop is a continuous map from a circle into a space. If f is a loop in X and if h is continuous then hf is a continuous map from the circle into Y and is therefore a loop.

The fundamental group is constructed from loops that begin and end at a fixed given point, the so called base point. It h preserves base points then a loop at the base point in X will be mapped to a loop at the base point in Y.

One can consistently define where a loop,f, begins and ends by thinking of it as a map of the unit interval into a space whose value is the same at 0 and 1. Then by definition the loop begins and ends at f(0),

 

[Math Amateur]

Thanks Lavinia

OK so after we are given that base point is preserved, it is continuity that ensures we have a loop in Y

Thanks again

Peter

 

[lavinia]

Thanks Lavinia

OK so after we are given that base point is preserved, it is continuity that ensures we have a loop in Y

Thanks again

Peter

right