8.6 The Freudenthal suspension theorem

8.6 The Freudenthal suspension theorem

Before proving the

Freudenthal suspension theorem

, we need some auxiliary lemmas about

connectedness

.
In

\autoref

cha:hlevels we proved a number of facts about

n

-

connected

maps and

n

-types for fixed

n

; here we are now interested in what happens when we vary

n

.
For

instance

, in

\autoref

prop:nconnected_tested_by_lv_n_dependent types we showed that

n

-connected maps are characterized by an “

induction

principle” relative to families of

n

-types.
If we want to “induct along” an

n

-connected map into a family of

k

-types for

k

>

n

, we don’t immediately know that there is a

function

by such an induction principle, but the following lemma says that at least our ignorance can be quantified.

Lemma 8.6.1

.

If

f

:

A

→

B

is

n

-connected and

P

:

B

→

k

⁢

-

⁢

Type

is a family of

k

-types for

k

≥

n

, then the induced function

(

–

∘

f

)

:

(

∏

b

:

B

P

⁢

(

b

)

)

→

(

∏

a

:

A

P

⁢

(

f

⁢

(

a

)

)

)

is

(

k

-

n

-

2

)

-truncated.

Proof.

We induct on the

natural number

k

-

n

.
When

k

=

n

, this is

\autoref

prop:nconnected_tested_by_lv_n_dependent types.
For the inductive step, suppose

f

is

n

-connected and

P

is a family of

k

+

1

-types.
To show that

(

–

∘

f

)

is

(

k

-

n

-

1

)

-truncated, let

k

:

∏

(

a

:

A

)

P

⁢

(

a

)

; then we have

𝖿𝗂𝖻

(

–

∘

f

)

⁢

(

k

)

≃

∑

(

g

:

∏

(

b

:

B

)

P

(

b

)

)

∏

(

a

:

A

)

g

⁢

(

f

⁢

(

a

)

)

=

k

⁢

(

a

)

.

Let

(

g

,

p

)

and

(

h

,

q

)

lie in this type, so

p

:

g

∘

f

∼

k

and

q

:

h

∘

f

∼

k

; then we also have

(

(

g

,

p

)

=

(

h

,

q

)

)

≃

(

∑

r

:

g

∼

h

r

∘

f

=

p

\centerdot

q

-

1

)

.

However, here the right-hand side is a fiber of the map

(

–

∘

f

)

:

(

∏

b

:

B

Q

⁢

(

b

)

)

→

(

∏

a

:

A

Q

⁢

(

f

⁢

(

a

)

)

)

where

Q

(

b

)

:

≡

(

g

(

b

)

=

h

(

b

)

)

.
Since

P

is a family of

(

k

+

1

)

-types,

Q

is a family of

k

-types, so the

inductive hypothesis

implies that this fiber is a

(

k

-

n

-

2

)

-type.
Thus, all

path spaces

of

𝖿𝗂𝖻

(

–

∘

f

)

⁢

(

k

)

are

(

k

-

n

-

2

)

-types, so it is a

(

k

-

n

-

1

)

-type.
∎

Recall that if

(

A

,

a

0

)

and

(

B

,

b

0

)

are pointed types, then
their

wedge

A

∨

B

is defined to be the

pushout

of

A

←

a

0

𝟏

→

b

0

B

.
There is a

canonical map

i

:

A

∨

B

→

A

×

B

defined by the two maps

λ

⁢

a

.

(

a

,

b

0

)

and

λ

⁢

b

.

(

a

0

,

b

)

; the following lemma essentially says that this map is highly connected if

A

and

B

are so.
It is a bit more convenient both to prove and use, however, if we use the

characterization

of connectedness from

\autoref

prop:nconnected_tested_by_lv_n_dependent types and substitute in the

universal property

of the wedge (generalized to type families).

Lemma 8.6.2

(Wedge connectivity lemma).

Suppose that

(

A

,

a

0

)

and

(

B

,

b

0

)

are

n

- and

m

-connected pointed types, respectively, with

n

,

m

≥

0

, and let

P

:

A

→

B

→

(

n

+

m

)

⁢

-

⁢

Type

.

Then for any

f

:

∏

(

a

:

A

)

P

⁢

(

a

,

b

0

)

and

g

:

∏

(

b

:

B

)

P

⁢

(

a

0

,

b

)

with

p

:

f

⁢

(

a

0

)

=

g

⁢

(

b

0

)

, there exists

h

:

∏

(

a

:

A

)

∏

(

b

:

B

)

P

⁢

(

a

,

b

)

with

homotopies

q

:

∏

a

:

A

h

⁢

(

a

,

b

0

)

=

f

⁢

(

a

)

𝑎𝑛𝑑

r

:

∏

b

:

B

h

⁢

(

a

0

,

b

)

=

g

⁢

(

b

)

such that

p

=

q

⁢

(

a

0

)

-

1

⁢

\centerdot

⁢

r

⁢

(

b

0

)

.

Proof.

Define

P

:

A

→

𝒰

by

P

(

a

)

:

≡

∑

k

:

∏

(

b

:

B

)

P

⁢

(

a

,

b

)

(

f

(

a

)

=

k

(

b

0

)

)

.

Then we have

(

g

,

p

)

:

P

⁢

(

a

0

)

.
Since

a

0

:

𝟏

→

A

is

(

n

-

1

)

-connected, if

P

is a family of

(

n

-

1

)

-types then we will have

ℓ

:

∏

(

a

:

A

)

P

⁢

(

a

)

such that

ℓ

⁢

(

a

0

)

=

(

g

,

p

)

, in which case we can define

h

(

a

,

b

)

:

≡

𝗉𝗋

1

(

ℓ

(

a

)

)

(

b

)

.
However, for fixed

a

, the type

P

⁢

(

a

)

is the fiber over

f

⁢

(

a

)

of the map

(

∏

b

:

B

P

⁢

(

a

,

b

)

)

→

P

⁢

(

a

,

b

0

)

given by precomposition with

b

0

:

𝟏

→

B

.
Since

b

0

:

𝟏

→

B

is

(

m

-

1

)

-connected, for this fiber to be

(

n

-

1

)

-connected, by

\autoref

thm:conn-trunc-variable-ind it suffices for each type

P

⁢

(

a

,

b

)

to be an

(

n

+

m

)

-type, which we have assumed.
∎

Let

(

X

,

x

0

)

be a pointed type, and recall the

definition

of the

suspension

Σ

⁢

X

from

\autoref

sec:suspension, with constructors

𝖭

,

𝖲

:

Σ

⁢

X

and

𝗆𝖾𝗋𝗂𝖽

:

X

→

(

𝖭

=

𝖲

)

.
We regard

Σ

⁢

X

as a

pointed space

with

basepoint

𝖭

, so that we have

Ω

Σ

X

:

≡

(

𝖭

=

Σ

⁢

X

𝖭

)

.
Then there is a canonical map

σ

:

X

→

Ω

⁢

Σ

⁢

X

σ

⁢

(

x

)

:

≡

𝗆𝖾𝗋𝗂𝖽

(

x

)

\centerdot

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

.

Remark 8.6.3

.

In classical

algebraic topology

, one considers the

reduced suspension

, in which the path

merid

⁢

(

x

0

)

is collapsed down to a point, identifying

N

and

S

.
The

reduced

and unreduced suspensions are

homotopy equivalent

, so the distinction is invisible to our purely homotopy-theoretic eyes — and

higher inductive types

only allow us to “identify” points up to a higher path anyway, there is no purpose to considering reduced suspensions in

homotopy type theory

.
However, the “unreducedness” of our suspension is the reason for the (possibly unexpected) appearance of

merid

⁢

(

x

0

)

-

1

in the definition of

σ

.

Our goal is now to prove the following.

Theorem 8.6.4

(The Freudenthal suspension theorem).

Suppose that

X

is

n

-connected and pointed, with

n

≥

0

.
Then the map

σ

:

X

→

Ω

⁢

Σ

⁢

(

X

)

is

2

⁢

n

-connected.

We will use the encode-decode method, but applied in a slightly different way.
In most cases so far, we have used it to characterize the

loop space

Ω

⁢

(

A

,

a

0

)

of some type as

equivalent

to some other type

B

, by constructing a family

𝖼𝗈𝖽𝖾

:

A

→

𝒰

with

𝖼𝗈𝖽𝖾

(

a

0

)

:

≡

B

and a family of

equivalences

𝖽𝖾𝖼𝗈𝖽𝖾

:

∏

(

x

:

A

)

𝖼𝗈𝖽𝖾

(

x

)

≃

(

a

0

=

x

)

.

In this case, however, we want to show that

σ

:

X

→

Ω

⁢

Σ

⁢

X

is

2

⁢

n

-connected.
We could use a truncated version of the previous method, such as we will see in

\autoref

sec:van-kampen, to prove that

∥

X

∥

2

⁢

n

→

∥

Ω

Σ

X

∥

2

⁢

n

is an equivalence—but this is a slightly weaker statement than the map being

2

⁢

n

-connected (see

\autoref

thm:conn-pik,

\autoref

thm:pik-conn).
However, note that in the general case, to prove that

𝖽𝖾𝖼𝗈𝖽𝖾

⁢

(

x

)

is an equivalence, we could equivalently be proving that its fibers are

contractible

, and we would still be able to use induction over the base type.
This we can generalize to prove connectedness of a map into a loop space, i.e. that the

truncations

of its fibers are contractible.
Moreover, instead of constructing

𝖼𝗈𝖽𝖾

and

𝖽𝖾𝖼𝗈𝖽𝖾

separately, we can

construct

directly a family of

codes for the truncations of the fibers

.

Definition 8.6.5

.

If

X

is

n

-connected and pointed with

n

≥

0

, then there is a family

𝖼𝗈𝖽𝖾

:

∏

y

:

Σ

⁢

X

(

𝖭

=

y

)

→

𝒰

(8.6.5)

such that

𝖼𝗈𝖽𝖾

⁢

(

𝖭

,

p

)

:

≡

∥

𝖿𝗂𝖻

σ

(

p

)

∥

2

⁢

n

≡

∥

∑

(

x

:

X

)

(

𝗆𝖾𝗋𝗂𝖽

(

x

)

\centerdot

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

=

p

)

∥

2

⁢

n

(8.6.6)

𝖼𝗈𝖽𝖾

⁢

(

𝖲

,

q

)

:

≡

∥

𝖿𝗂𝖻

𝗆𝖾𝗋𝗂𝖽

(

q

)

∥

2

⁢

n

≡

∥

∑

(

x

:

X

)

(

𝗆𝖾𝗋𝗂𝖽

(

x

)

=

q

)

∥

2

⁢

n

.

(8.6.6)

Our eventual goal will be to prove that

𝖼𝗈𝖽𝖾

⁢

(

y

,

p

)

is contractible for all

y

:

Σ

⁢

X

and

p

:

𝖭

=

y

.
Applying this with

y

:

≡

𝖭

will show that all fibers of

σ

are

2

⁢

n

-connected, and thus

σ

is

2

⁢

n

-connected.

Proof of

\autoref

thm:freudcode.

We define

𝖼𝗈𝖽𝖾

⁢

(

y

,

p

)

by induction on

y

:

Σ

⁢

X

, where the first two cases are (

8.6.6

) and (

8.6.6

).
It remains to construct, for each

x

1

:

X

, a dependent path

𝖼𝗈𝖽𝖾

⁢

(

𝖭

)

=

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

λ

y

.

(

𝖭

=

y

)

→

𝒰

𝖼𝗈𝖽𝖾

⁢

(

𝖲

)

.

By

\autoref

thm:dpath-arrow, this is equivalent to giving a family of paths

∏

q

:

𝖭

=

𝖲

𝖼𝗈𝖽𝖾

⁢

(

𝖭

)

⁢

(

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

λ

y

.

(

𝖭

=

y

)

⁢

(

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

-

1

,

q

)

)

=

𝖼𝗈𝖽𝖾

⁢

(

𝖲

)

⁢

(

q

)

.

And by univalence and transport in path types, this is equivalent to a family of equivalences

∏

q

:

𝖭

=

𝖲

𝖼𝗈𝖽𝖾

⁢

(

𝖭

,

q

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

-

1

)

≃

𝖼𝗈𝖽𝖾

⁢

(

𝖲

,

q

)

.

We will define a family of maps

∏

q

:

𝖭

=

𝖲

𝖼𝗈𝖽𝖾

⁢

(

𝖭

,

q

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

-

1

)

→

𝖼𝗈𝖽𝖾

⁢

(

𝖲

,

q

)

.

(8.6.7)

and then show that they are all equivalences.
Thus, let

q

:

𝖭

=

𝖲

; by the universal property of truncation and the definitions of

𝖼𝗈𝖽𝖾

⁢

(

𝖭

,

–

)

and

𝖼𝗈𝖽𝖾

⁢

(

𝖲

,

–

)

, it will suffice to define for each

x

2

:

X

, a map

(

𝗆𝖾𝗋𝗂𝖽

(

x

2

)

\centerdot

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

=

q

\centerdot

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

-

1

)

→

∥

∑

(

x

:

X

)

(

𝗆𝖾𝗋𝗂𝖽

(

x

)

=

q

)

∥

2

⁢

n

.

Now for each

x

1

,

x

2

:

X

, this type is

2

⁢

n

-truncated, while

X

is

n

-connected.
Thus, by

\autoref

thm:wedge-connectivity, it suffices to define this map when

x

1

is

x

0

, when

x

2

is

x

0

, and check that they agree when both are

x

0

.

When

x

1

is

x

0

, the

hypothesis

is

r

:

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

2

)

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

=

q

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

.
Thus, by canceling

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

from

r

to get

r

′

:

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

2

)

=

q

, so we can define the image to be

|

(

x

2

,

r

′

)

|

2

⁢

n

.

When

x

2

is

x

0

, the hypothesis is

r

:

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

=

q

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

-

1

.
Rearranging this, we obtain

r

′′

:

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

=

q

, and we can define the image to be

|

(

x

1

,

r

′′

)

|

2

⁢

n

.

Finally, when both

x

1

and

x

2

are

x

0

, it suffices to show the resulting

r

′

and

r

′′

agree; this is an easy lemma about path composition.
This

completes

the definition of (

8.6.7

).
To show that it is a family of equivalences, since being an equivalence is a mere proposition and

x

0

:

𝟏

→

X

is (at least)

(

-

1

)

-connected, it suffices to assume

x

1

is

x

0

.
In this case, inspecting the above construction we see that it is essentially the

2

⁢

n

-truncation of the function that cancels

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

, which is an equivalence.
∎

In

addition

to (

8.6.6

) and (

8.6.6

), we will need to extract from the construction of

𝖼𝗈𝖽𝖾

some information about how it acts on paths.
For this we use the following lemma.

Lemma 8.6.8

.

Let

A

:

U

,

B

:

A

→

U

, and

C

:

∏

(

a

:

A

)

B

⁢

(

a

)

→

U

, and also

a

1

,

a

2

:

A

with

m

:

a

1

=

a

2

and

b

:

B

⁢

(

a

2

)

.
Then the function

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

C

^

⁢

(

𝗉𝖺𝗂𝗋

=

⁢

(

m

,

t

)

,

–

)

:

C

⁢

(

a

1

,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

B

⁢

(

m

-

1

,

b

)

)

→

C

⁢

(

a

2

,

b

)

,

where

t

:

transport

B

⁢

(

m

,

transport

B

⁢

(

m

-

1

,

b

)

)

=

b

is the

obvious

coherence path and

C

^

:

(

∑

(

a

:

A

)

B

⁢

(

a

)

)

→

U

is the uncurried form of

C

, is equal to the equivalence obtained by univalence from the composite

C

⁢

(

a

1

,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

B

⁢

(

m

-

1

,

b

)

)

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

λ

⁢

a

.

B

⁢

(

a

)

→

𝒰

⁢

(

m

,

C

⁢

(

a

1

)

)

⁢

(

b

)

(by \eqref{eq:transport-arrow})

=

C

⁢

(

a

2

,

b

)

.

(by $\mathsf{happly}(\mathsf{apd}_{C}\mathopen{}\left(m\right)\mathclose{},b)$)

Proof.

By path induction, we may assume

a

2

is

a

1

and

m

is

𝗋𝖾𝖿𝗅

a

1

, in which case both functions are the

identity

.
∎

We apply this lemma with

A

:

≡

Σ

X

and

B

:

≡

λ

y

.

(

𝖭

=

y

)

and

C

:

≡

𝖼𝗈𝖽𝖾

, while

a

1

:

≡

𝖭

and

a

2

:

≡

𝖲

and

m

:

≡

𝗆𝖾𝗋𝗂𝖽

(

x

1

)

for some

x

1

:

X

, and finally

b

:

≡

q

is some path

𝖭

=

𝖲

.
The computation rule for induction over

Σ

⁢

X

identifies

𝖺𝗉𝖽

C

(

m

)

with a path constructed in a certain way out of univalence and function extensionality.
The second function described in

\autoref

thm:freudlemma essentially consists of undoing these applications of univalence and function extensionality, reducing back to the particular functions (

8.6.7

) that we defined using

\autoref

thm:wedge-connectivity.
Therefore,

\autoref

thm:freudlemma says that transporting along

𝗉𝖺𝗂𝗋

=

⁢

(

q

,

t

)

essentially recovers these functions.

Finally, by construction, when

x

1

or

x

2

coincides with

x

0

and the input is in the image of

|

–

|

2

⁢

n

, we know more explicitly what these functions are.
Thus, for any

x

2

:

X

, we have

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖼𝗈𝖽𝖾

^

(

𝗉𝖺𝗂𝗋

=

(

𝗆𝖾𝗋𝗂𝖽

(

x

0

)

,

t

)

,

|

(

x

2

,

r

)

|

2

⁢

n

)

=

|

(

x

1

,

r

′

)

|

2

⁢

n

(8.6.9)

where

r

:

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

2

)

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

B

⁢

(

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

,

q

)

is arbitrary as before, and

r

′

:

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

2

)

=

q

is obtained from

r

by identifying its

end point

with

q

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

and canceling

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

.
Similarly, for any

x

1

:

X

, we have

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖼𝗈𝖽𝖾

^

(

𝗉𝖺𝗂𝗋

=

(

𝗆𝖾𝗋𝗂𝖽

(

x

1

)

,

t

)

,

|

(

x

0

,

r

)

|

2

⁢

n

)

=

|

(

x

1

,

r

′′

)

|

2

⁢

n

(8.6.10)

where

r

:

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

B

⁢

(

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

-

1

,

q

)

, and

r

′′

:

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

=

q

is obtained by identifying its end point and rearranging paths.

Proof of

\autoref

thm:freudenthal.

It remains to show that

𝖼𝗈𝖽𝖾

⁢

(

y

,

p

)

is contractible for each

y

:

Σ

⁢

X

and

p

:

𝖭

=

y

.
First we must choose a center of

contraction

, say

c

⁢

(

y

,

p

)

:

𝖼𝗈𝖽𝖾

⁢

(

y

,

p

)

.
This corresponds to the definition of the function

𝖾𝗇𝖼𝗈𝖽𝖾

in our previous proofs, so we define it by transport.
Note that in the special case when

y

is

𝖭

and

p

is

𝗋𝖾𝖿𝗅

𝖭

, we have

𝖼𝗈𝖽𝖾

(

𝖭

,

𝗋𝖾𝖿𝗅

𝖭

)

≡

∥

∑

(

x

:

X

)

(

𝗆𝖾𝗋𝗂𝖽

(

x

)

\centerdot

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

=

𝗋𝖾𝖿𝗅

𝖭

)

∥

2

⁢

n

.

Thus, we can choose

c

(

𝖭

,

𝗋𝖾𝖿𝗅

𝖭

)

:

≡

|

(

x

0

,

𝗋𝗂𝗇𝗏

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

)

|

2

⁢

n

, where

rinv

q

is the obvious path

q

⁢

\centerdot

⁢

q

-

1

=

𝗋𝖾𝖿𝗅

for any

q

.
We can now obtain

c

:

∏

(

y

:

Σ

X

)

∏

(

p

:

𝖭

=

y

)

𝖼𝗈𝖽𝖾

⁢

(

y

,

p

)

by path induction on

p

, but it will be important below that we can also give a concrete definition in terms of transport:

c

(

y

,

p

)

:

≡

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖼𝗈𝖽𝖾

^

(

𝗉𝖺𝗂𝗋

=

(

p

,

𝗍𝗂𝖽

p

)

,

c

(

𝖭

,

𝗋𝖾𝖿𝗅

𝖭

)

)

where

𝖼𝗈𝖽𝖾

^

:

(

∑

(

y

:

Σ

X

)

(

𝖭

=

y

)

)

→

𝒰

is the uncurried version of

𝖼𝗈𝖽𝖾

, and

𝗍𝗂𝖽

p

:

p

*

(

𝗋𝖾𝖿𝗅

)

=

p

is a standard lemma.

Next, we must show that every element of

𝖼𝗈𝖽𝖾

⁢

(

y

,

p

)

is equal to

c

⁢

(

y

,

p

)

.
Again, by path induction, it suffices to assume

y

is

𝖭

and

p

is

𝗋𝖾𝖿𝗅

𝖭

.
In fact, we will prove it more generally when

y

is

𝖭

and

p

is arbitrary.
That is, we will show that for any

p

:

𝖭

=

𝖭

and

d

:

𝖼𝗈𝖽𝖾

⁢

(

𝖭

,

p

)

we have

d

=

c

⁢

(

𝖭

,

p

)

.
Since this equality is a

(

2

⁢

n

-

1

)

-type, we may assume

d

is of the form

|

(

x

1

,

r

)

|

2

⁢

n

for some

x

1

:

X

and

r

:

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

=

p

.

Now by a further path induction, we may assume that

r

is

reflexivity

, and

p

is

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

.
(This is why we generalized to arbitrary

p

above.)
Thus, we have to prove that

|

(

x

1

,

𝗋𝖾𝖿𝗅

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

)

|

2

⁢

n

=

c

(

𝖭

,

𝗋𝖾𝖿𝗅

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

1

)

⁢

\centerdot

⁢

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

)

.

(8.6.11)

By definition, the right-hand side of this equality is

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖼𝗈𝖽𝖾

^

(

𝗉𝖺𝗂𝗋

=

(

𝗆𝖾𝗋𝗂𝖽

(

x

1

)

\centerdot

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

,

¯

)

,

|

(

x

0

,

¯

)

|

2

⁢

n

)

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖼𝗈𝖽𝖾

^

⁢

(

𝗉𝖺𝗂𝗋

=

⁢

(

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

,

¯

)

,

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖼𝗈𝖽𝖾

^

(

𝗉𝖺𝗂𝗋

=

(

𝗆𝖾𝗋𝗂𝖽

(

x

1

)

,

¯

)

,

|

(

x

0

,

¯

)

|

2

⁢

n

)

)

=

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

𝖼𝗈𝖽𝖾

^

(

𝗉𝖺𝗂𝗋

=

(

𝗆𝖾𝗋𝗂𝖽

⁢

(

x

0

)

-

1

,

¯

)

,

|

(

x

1

,

¯

)

|

2

⁢

n

)

=

|

(

x

1

,

¯

)

|

2

⁢

n

where the underscore

¯

ought to be filled in with suitable coherence paths.
Here the first step is functoriality of transport, the second invokes (

8.6.10

), and the third invokes (

8.6.9

) (with transport moved to the other side).
Thus we have the same first component as the left-hand side of (

8.6.11

).
We leave it to the reader to verify that the coherence paths all cancel, giving reflexivity in the second component.
∎

Corollary 8.6.12

(Freudenthal Equivalence).

Suppose that

X

is

n

-connected and pointed, with

n

≥

0

.
Then

∥

X

∥

2

⁢

n

≃

∥

Ω

Σ

(

X

)

∥

2

⁢

n

.

Proof.

By

\cref

thm:freudenthal,

σ

is

2

⁢

n

-connected. By

\cref

lem:connected-map-equiv-truncation, it is therefore an
equivalence on

2

⁢

n

-truncations.
∎

One important

corollary

of the Freudenthal suspension theorem is that the homotopy groups of
spheres are stable in a certain range (these are the northeast-to-southwest

diagonals

in

\autoref

tab:homotopy-groups-of-spheres):

Corollary 8.6.13

(Stability for Spheres).

If

k

≤

2

⁢

n

-

2

, then

π

k

+

1

⁢

(

S

n

+

1

)

=

π

k

⁢

(

S

n

)

.

Proof.

Assume

k

≤

2

⁢

n

-

2

.
By

\cref

cor:sn-connected,

𝕊

n

is

(

n

-

1

)

-connected. Therefore,
by

\cref

cor:freudenthal-equiv,

∥

Ω

(

Σ

(

𝕊

n

)

)

∥

2

⁢

(

n

-

1

)

=

∥

𝕊

n

∥

2

⁢

(

n

-

1

)

.

By

\cref

lem:truncation-le, because

k

≤

2

⁢

(

n

-

1

)

, applying

∥

–

∥

k

to both sides shows that this equation holds for

k

:

∥

Ω

(

Σ

(

𝕊

n

)

)

∥

k

=

∥

𝕊

n

∥

k

.

(8.6.14)

Then, the main idea of the proof is as follows; we omit checking that these
equivalences act appropriately on the base points of these spaces:

π

k

+

1

⁢

(

𝕊

n

+

1

)

≡

∥

Ω

k

+

1

(

𝕊

n

+

1

)

∥

0

≡

∥

Ω

k

(

Ω

(

𝕊

n

+

1

)

)

∥

0

≡

∥

Ω

k

(

Ω

(

Σ

(

𝕊

n

)

)

)

∥

0

=

Ω

k

(

∥

(

Ω

(

Σ

(

𝕊

n

)

)

)

∥

k

)

(by \autorefthm:path-truncation)

=

Ω

k

(

∥

𝕊

n

∥

k

)

(by \eqref{eq:freudenthal-for-spheres})

=

∥

Ω

k

(

𝕊

n

)

∥

0

(by \autorefthm:path-truncation)

≡

π

k

⁢

(

𝕊

n

)

.

∎

This means that once we have calculated one entry in one of these stable
diagonals, we know all of them. For example:

Theorem 8.6.15

.

π

n

⁢

(

𝕊

n

)

=

ℤ

for every

n

≥

1

.

Proof.

The proof is by induction on

n

. We already have

π

1

⁢

(

𝕊

1

)

=

ℤ

(

\autoref

cor:pi1s1) and

π

2

⁢

(

𝕊

2

)

=

ℤ

(

\autoref

cor:pis2-hopf).
When

n

≥

2

,

n

≤

(

2

⁢

n

-

2

)

. Therefore, by

\cref

cor:stability-spheres,

π

n

+

1

⁢

(

S

n

+

1

)

=

π

n

⁢

(

S

n

)

, and
this equivalence, combined with the inductive hypothesis, gives the result.
∎

Corollary 8.6.16

.

𝕊

n

+

1

is not an

n

-type for any

n

≥

-

1

.

Title

8.6 The Freudenthal suspension theorem

\metatable

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