# 1.03 Concatenation and the fundamental group

## Video

Below the video you will find accompanying notes and some pre-class questions.

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## Notes

### Concatenation

*(0.10)* Let \(X\) be a topological space and suppose
\(\alpha,\beta\colon[0,1]\to X\) are paths such that
\(\alpha(1)=\beta(0)\). The *concatenation* \(\beta\cdot\alpha\) is
the path \[(\beta\cdot\alpha)(t)=\begin{cases}\alpha(2t)&\mbox{ if
}t\in[0,1/2]\\\beta(2t-1)&\mbox{ if }t\in[1/2,1].\end{cases}\]

### The fundamental group

*(3.49)* Given a topological space \(X\) and a basepoint \(x\in X\),
write \(\Omega_xX\) for the set of all loops in \(X\) based at
\(x\). The relation \(\simeq\) that two loops
\(\alpha,\beta\in\Omega_xX\) are based-homotopic is an *equivalence
relation*. We write \(\pi_1(X,x)\) for the set of equivalence
classes \(\Omega_xX/\simeq\) of loops up to based homotopy.

We can make \(\pi_1(X,x)\) into a group under concatenation: if \([\alpha]\in\pi_1(X,x)\) denotes the homotopy class of the loop \(\alpha\) then \[[\beta]\cdot[\alpha]=[\beta\cdot\alpha].\]

The operation \([\beta]\cdot[\alpha]=[\beta\cdot\alpha]\) defines a group structure on \(\pi_1(X,x)\). In particular:

- it is well-defined on homotopy classes,
- it is associative,
- it has an identity (the identity element is the constant loop \(\epsilon(t)=x\)),
- each homotopy class of loops \([\gamma]\) has an inverse \([\bar{\gamma}]\), where \(\bar{\gamma}(t)=\gamma(1-t)\) is the loop which runs around \(\gamma\) in reverse.

*(8.30)* We will show that the constant loop \(\epsilon\) is an
identity for concatenation on \(\pi_1(X,x)\); the other parts are
left as exercises.

The concatenation \(\gamma\cdot\epsilon\) stays at \(x\) for time
\(1/2\) and then moves around \(\gamma\) at double-speed. This is
not *equal to* the loop \(\gamma\), but it only differs in the way
it is parametrised. By playing with the parametrisation, we can
construct a homotopy \(\gamma_s\) from this concatenation to the
original loop \(\gamma\). \[ \gamma_s(t)=\begin{cases}
\epsilon(t)&\mbox{ if }t\leq 1/2(1-s)\\ \gamma((2-s)t+s-1)&\mbox{ if
}t\geq 1/2(1-s) \end{cases} \] *(10.46)* To understand this formula,
observe that at the stage \(s\) in the homotopy, the loop
\(\gamma_s\) stays at \(x\) for time \(1/2(1-s)\) (\(1/2\) when
\(s=0\) and \(0\) when \(s=1\)) then moves around \(\gamma\) at
\((2-s)\) times the speed of the original loop (twice as fast when
\(s=0\), the same speed as the original when \(s=1\)). Here is a
picture of the domain of the homotopy:

*(14.46)* This homotopy is continuous by one of the exercises we
will do in class when we learn about topological spaces; it is a
homotopy rel endpoints since \(\gamma_s(0)=\gamma_s(1)=x\) for all
\(s\); and it satisfies \(\gamma_0=\gamma\cdot\epsilon\) and
\(\gamma_1=\gamma\). A similar homotopy works for
\(\epsilon\cdot\gamma\).

## Pre-class questions

- Suppose that \(\alpha_t\) is a homotopy between \(\alpha_0\) and \(\alpha_1\) and \(\beta_t\) is a homotopy between \(\beta_0\) and \(\beta_1\). Can you write down a homotopy between \(\beta_0\cdot\alpha_0\) and \(\beta_1\cdot\alpha_1\)? This verifies one of the claims from the lemma in the video/notes: which claim?

## Navigation

- Previous video:
**1.02 Paths, loops, and homotopies**. - Next video:
**1.04 Examples and simply-connectedness**. - Index of all lectures.