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 [itex] h: X \rightarrow Y [/itex] is a continuous map that carries the point [itex] x_0 [/itex] of X to the point [itex] y_0 [/itex] of Y.

We denote this fact by writing:

[itex] h: ( X, x_0) \rightarrow (Y, y_0) [/itex]

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

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

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


i.e. [itex] h \circ f (i^{'} ) [/itex] [itex] = y^' [/itex]

BUT

[itex] h \circ f (i^{'} ) = h(f(i^{'} )) = h(x^{'} ) [/itex]

But (see above) we only know of h that it maps [itex] x_0 [/itex] into [itex] y_0 [/itex]? {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

 

[blue_raver22]

,

Thank you for your question. Algebraic topology is a complex subject, so I will do my best to explain the mechanics of this concept in a clear and concise manner.

First, let's define some terms. A loop in a space X is a continuous map f: I \rightarrow X, where I is the unit interval [0,1]. This means that for every point in the interval, there is a corresponding point in X. The point x_0 in X is the basepoint of the loop, which is the starting and ending point of the loop. Now, let's consider a continuous map h: X \rightarrow Y, where X and Y are topological spaces. This means that for every point in X, there is a corresponding point in Y. We denote this fact by writing h: ( X, x_0) \rightarrow (Y, y_0), where x_0 is the basepoint of X and y_0 is the basepoint of Y.

Now, let's consider the composite map h \circ f: I \rightarrow Y. This means that for every point in the interval I, there is a corresponding point in Y. We can also think of this map as h(f(t)), where t is a point in the interval I. Since f is a loop based at x_0, we know that f(0) = f(1) = x_0. Therefore, h(f(0)) = h(x_0) = y_0 and h(f(1)) = h(x_0) = y_0. This means that h \circ f is also a loop based at y_0 in Y.

To address your confusion, let's consider your example. If we have a point i^{'} \in [0,1] that is mapped by f into x^', then h \circ f (i^{'}) = h(f(i^{'})) = h(x^{'}) = y^'. This is because h maps every point in X to a corresponding point in Y, including the basepoint x_0 to the basepoint y_0. Therefore, we don't need to know any additional information about h to show that h \circ f is a loop based at y_0.

I hope this clarifies the mechanics of this concept for you. Please let me know if you have any further questions.