\(\newcommand{\from}{\colon}\)\(\newcommand{\cross}{\times}\)\(\newcommand{\cover}[1]{{\widetilde{#1}}}\)Questions
handed in by students on 2017-10-18.

General - 

1. How does one define a function on a graph?

Ans: Suppose that \(X\) is a space and \(Y = X/{\sim}\) is a quotient
of \(X\).  We can obtain a function on \(Y\) by finding a function on
\(X\) which is constant on equivalence classes.

TL;DR - Define the function on the edges of the graph, and check that
it is consistent across vertices.

2. Is there an intuitive way to think of covering spaces?

Ans: A covering space \( \cover{X} \) is a way of "unwrapping" the
base space \( X \).  Or, conversely, you could think of the covering
map \( p \) as "winding" \( \cover{X} \) around \( X \).  To visualise
how the circle covers the circle, it may help to find some rubber
bands and wind them around things...

Homework -

1. On Exercise Sheet 2, what sort of assumptions are you looking for
in writing the homotopy classes of the alphabet?

Ans: Do you mean "justifications" when you write "assumptions"?  Some
justification is definitely needed.  You should not give \(26^2\)
explanations, however.  Think about how to neatly organise the work.

Above and beyond -

1. Is there \( \pi_n(X, x_0) \) for \( n > 1 \)?  If so, what is it?

Ans: These are the higher homotopy groups.  Instead of loops, we use
\(n\)-spheres.  The concatenation operation is a bit more involved.
See chapter four of Hatcher's book for a discussion.