8.5.2 The Hopf construction

Definition 8.5.1.

An H-spaceconsists of

• a type $A$,

• a base point $e:A$,

• a binary operation $\mu :A\times A\to A$, and

• for every $a:A$, equalities $\mu (e,a)=a$ and $\mu (a,e)=a$.

Lemma 8.5.2.

Let $A$ be a connected H-space. Then for every $a\mathrm{:}A$, the maps $\mu \mathit{}\mathrm{(}a\mathrm{,}\mathit{\text{–}}\mathrm{)}\mathrm{:}A\mathrm{\to }A$ and $\mu \mathit{}\mathrm{(}\mathit{\text{–}}\mathrm{,}a\mathrm{)}\mathrm{:}A\mathrm{\to }A$ are equivalences.

Proof.

Let us prove that for every $a:A$ the map $\mu (a,\text{–})$ is an equivalence. Theother statement is symmetric.The statement that $\mu (a,\text{–})$ is an equivalence corresponds to a type family$P:A\to \mathrm{𝖯𝗋𝗈𝗉}$ and proving it corresponds to finding a section of this typefamily.

The type $\mathrm{𝖯𝗋𝗈𝗉}$ is a set (thm:hleveln-of-hlevelSn) hence we candefine a new type family $P^{\prime }:\parallel A{\parallel }_0\to \mathrm{𝖯𝗋𝗈𝗉}$ by $P^{\prime }(|a{|}_0):\equiv P(a)$. But $A$ is connected by assumption, hence ${\parallel A\parallel }_0$ iscontractible. This implies that in order to find a section of $P^{\prime }$, it isenough to find a point in the fiber of $P^{\prime }$ over ${\left|e\right|}_0$. But we have$P^{\prime }(|e{|}_0)=P(e)$ which is inhabited because $\mu (e,\text{–})$ is equal to theidentity map by definition of an H-space, hence is an equivalence.

We have proved that for every $x:\parallel A{\parallel }_0$ the proposition $P^{\prime }(x)$ is true,hence in particular for every $a:A$ the proposition $P(a)$ is true because$P(a)$ is $P^{\prime }(|a{|}_0)$.∎

Definition 8.5.3.

Let $A$ be a connected H-space. We define a fibration over $\mathrm{\Sigma }\mathit{}A$ usinglem:fibration-over-pushout.

Given that $\mathrm{\Sigma }\mathit{}A$ is the pushout $\mathrm{1}{\mathrm{\bigsqcup }}^A\mathrm{1}$, we can define afibration over $\mathrm{\Sigma }\mathit{}A$ by specifying

• two fibrations over $\mathrm{𝟏}$ (i.e. two types $F_1$ and $F_2$), and

• a family $e:A\to (F_1\simeq F_2)$ of equivalences between$F_1$ and $F_2$, one for every element of $A$.

We take $A$ for $F_{\mathrm{1}}$ and $F_{\mathrm{2}}$, and for $a\mathrm{:}A$ we take the equivalence$\mu \mathit{}\mathrm{(}a\mathrm{,}\mathit{\text{–}}\mathrm{)}$ for $e\mathit{}\mathrm{(}a\mathrm{)}$.

According to lem:fibration-over-pushout, we have the followingdiagram:

and the fibration we just constructed is a fibration over $\mathrm{\Sigma }A$ whose totalspace is the pushout of the top line.

Moreover, with $f(x,y):\equiv (\mu (x,y),y)$ we have the following diagram:

The diagram commutes and the three vertical maps are equivalences, the inverseof $f$ being the function $g$ defined by

$$g(u,v):\equiv (\mu {(\text{–},v)}^{-1}(u),v).$$

This shows that the two lines are equivalent (hence equal) spans, so the totalspace of the fibration we constructed is equivalent to the pushout of the bottomline.And by definition, this latter pushout is the join of $A$ with itself (see sec:colimits).We have proven:

Lemma 8.5.4.

Given a connected H-space $A$, there is a fibration, called theHopf construction,over $\mathrm{\Sigma }\mathit{}A$ with fiber $A$ and total space $A\mathrm{*}A$.