8.5.2 The Hopf construction

8.5.2 The Hopf construction

Definition 8.5.1

.

An

H-space

consists of

•

a type

A

,

•

a

base point

e

:

A

,

•

a

binary operation

μ

:

A

×

A

→

A

, and

•

for every

a

:

A

, equalities

μ

⁢

(

e

,

a

)

=

a

and

μ

⁢

(

a

,

e

)

=

a

.

Lemma 8.5.2

.

Let

A

be a connected H-space. Then for every

a

:

A

, the maps

μ

⁢

(

a

,

–

)

:

A

→

A

and

μ

⁢

(

–

,

a

)

:

A

→

A

are equivalences.

Proof.

Let us prove that for every

a

:

A

the map

μ

⁢

(

a

,

–

)

is an equivalence. The
other statement is

symmetric

.
The statement that

μ

⁢

(

a

,

–

)

is an equivalence corresponds to a type family

P

:

A

→

𝖯𝗋𝗈𝗉

and proving it corresponds to finding a

section

of this type
family.

The type

𝖯𝗋𝗈𝗉

is a set (

\autoref

thm:hleveln-of-hlevelSn) hence we can
define a new type family

P

′

:

∥

A

∥

0

→

𝖯𝗋𝗈𝗉

by

P

′

(

|

a

|

0

)

:

≡

P

(

a

)

. But

A

is connected by

assumption

, hence

∥

A

∥

0

is
contractible. This implies that in order to find a section of

P

′

, it is
enough to find a point in the fiber of

P

′

over

|

e

|

0

. But we have

P

′

(

|

e

|

0

)

=

P

(

e

)

which is inhabited because

μ

⁢

(

e

,

–

)

is equal to the

identity map

by

definition

of an H-space, hence is an equivalence.

We have proved that for every

x

:

∥

A

∥

0

the

proposition

P

′

⁢

(

x

)

is true,
hence in particular for every

a

:

A

the proposition

P

⁢

(

a

)

is true because

P

⁢

(

a

)

is

P

′

(

|

a

|

0

)

.
∎

Definition 8.5.3

.

Let

A

be a connected H-space. We define a fibration over

Σ

⁢

A

using

\autoref

lem:fibration-over-pushout.

Given that

Σ

⁢

A

is the pushout

1

⊔

A

1

, we can define a
fibration over

Σ

⁢

A

by specifying

•

two fibrations over

𝟏

(i.e. two types

F

1

and

F

2

), and

•

a family

e

:

A

→

(

F

1

≃

F

2

)

of equivalences between

F

1

and

F

2

, one for every element of

A

.

We take

A

for

F

1

and

F

2

, and for

a

:

A

we take the equivalence

μ

⁢

(

a

,

–

)

for

e

⁢

(

a

)

.

According to

\autoref

lem:fibration-over-pushout, we have the following
diagram:

\xymatrix

⁢

A

⁢

\ar

⁢

@

-

≫

[

d

]

⁢

&

⁢

A

×

A

⁢

\ar

⁢

[

l

]

-

⁢

𝗉𝗋

2

⁢

\ar

⁢

@

-

≫

𝗉𝗋

1

[

d

]

⁢

\ar

⁢

[

r

]

-

⁢

μ

⁢

&

⁢

A

⁢

\ar

⁢

@

-

≫

[

d

]

⁢

1

⁢

&

⁢

A

⁢

\ar

⁢

[

r

]

⁢

\ar

⁢

[

l

]

⁢

&

⁢

1

and the fibration we just constructed is a fibration over

Σ

⁢

A

whose total
space is the pushout of the top line.

Moreover, with

f

(

x

,

y

)

:

≡

(

μ

(

x

,

y

)

,

y

)

we have the following diagram:

\xymatrix

⁢

A

⁢

\ar

𝗂𝖽

⁢

&

d

⁢

A

×

A

⁢

\ar

⁢

[

l

]

-

⁢

𝗉𝗋

2

⁢

\ar

f

⁢

[

d

]

⁢

\ar

⁢

[

r

]

-

⁢

μ

⁢

&

⁢

A

⁢

\ar

d

𝗂𝖽

⁢

A

⁢

&

⁢

A

×

A

⁢

\ar

-

⁢

𝗉𝗋

2

⁢

[

l

]

⁢

\ar

-

⁢

𝗉𝗋

1

⁢

[

r

]

⁢

&

⁢

A

The diagram commutes and the three vertical maps are equivalences, the

inverse

of

f

being the

function

g

defined by

g

(

u

,

v

)

:

≡

(

μ

⁢

(

–

,

v

)

-

1

(

u

)

,

v

)

.

This shows that the two lines are

equivalent

(hence equal) spans, so the total
space of the fibration we constructed is equivalent to the pushout of the bottom
line.
And by definition, this latter pushout is the

join

of

A

with itself (see

\autoref

sec:colimits).
We have proven:

Lemma 8.5.4

.

Given a connected H-space

A

, there is a fibration, called the

Hopf construction

,
over

Σ

⁢

A

with fiber

A

and total space

A

*

A

.

Title

8.5.2 The Hopf construction

\metatable

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