8.2 Connectedness of suspensions

8.2 Connectedness of suspensions

Recall from

\autoref

sec:connectivity that a type

A

is called

n

-connected

if

∥

A

∥

n

is

contractible

.
The aim of this

section

is to prove that the

operation

of

suspension

from

\autoref

sec:suspension increases connectedness.

Theorem 8.2.1

.

If

A

is

n

-connected then the suspension of

A

is

(

n

+

1

)

-connected.

Proof.

We remarked in

\autoref

sec:

colimits

that the suspension of

A

is the

pushout

𝟏

⊔

A

𝟏

, so we need to
prove that the following type is contractible:

∥

𝟏

⊔

A

𝟏

∥

n

+

1

.

By

\autoref

reflectcommutespushout we know that

∥

𝟏

⊔

A

𝟏

∥

n

+

1

is a pushout in

(

n

+

1

)

⁢

-

⁢

𝖳𝗒𝗉𝖾

of the

diagram

\xymatrix

∥

A

∥

n

+

1

\ar

[

d

]

\ar

[

r

]

&

∥

𝟏

∥

n

+

1

∥

𝟏

∥

n

+

1

&

.

Given that

∥

𝟏

∥

n

+

1

=

𝟏

, the type

∥

𝟏

⊔

A

𝟏

∥

n

+

1

is also a pushout of the following diagram in

(

n

+

1

)

⁢

-

⁢

𝖳𝗒𝗉𝖾

(because both diagrams are equal)

𝒟

=

\xymatrix

∥

A

∥

_n+1

\ar

[d]

\ar

[r] &

𝟏

𝟏

&

.

We will now prove that

𝟏

is also a pushout of

𝒟

in

(

n

+

1

)

⁢

-

⁢

𝖳𝗒𝗉𝖾

.
Let

E

be an

(

n

+

1

)

-truncated type; we need to prove that the following map
is an

equivalence

{

(

𝟏

→

E

)

⟶

𝖼𝗈𝖼𝗈𝗇𝖾

𝒟

⁢

(

E

)

y

⟼

(

y

,

y

,

λ

⁢

u

.

𝗋𝖾𝖿𝗅

y

⁢

(

⋆

)

)

.

where we recall that

𝖼𝗈𝖼𝗈𝗇𝖾

𝒟

⁢

(

E

)

is the type

∑

(

f

:

𝟏

→

E

)

∑

(

g

:

𝟏

→

E

)

(

∥

A

∥

n

+

1

→

(

f

(

⋆

)

=

E

g

(

⋆

)

)

)

.

The map

{

(

𝟏

→

E

)

⟶

E

f

⟼

f

⁢

(

⋆

)

is an equivalence, hence
we also have

𝖼𝗈𝖼𝗈𝗇𝖾

𝒟

(

E

)

=

∑

(

x

:

E

)

∑

(

y

:

E

)

(

∥

A

∥

n

+

1

→

(

x

=

E

y

)

)

.

Now

A

is

n

-connected hence so is

∥

A

∥

n

+

1

because

∥

∥

A

∥

n

+

1

∥

n

=

∥

A

∥

n

=

𝟏

, and

(

x

=

E

y

)

is

n

-truncated because

E

is

(

n

+

1

)

-connected. Hence by

\autoref

connectedtotruncated the
following map is an equivalence

{

(

x

=

E

y

)

⟶

(

∥

A

∥

n

+

1

→

(

x

=

E

y

)

)

p

⟼

λ

⁢

z

.

p

Hence we have

𝖼𝗈𝖼𝗈𝗇𝖾

𝒟

(

E

)

=

∑

(

x

:

E

)

∑

(

y

:

E

)

(

x

=

E

y

)

.

But the following map is an equivalence

{

E

⟶

∑

(

x

:

E

)

∑

(

y

:

E

)

(

x

=

E

y

)

x

⟼

(

x

,

x

,

𝗋𝖾𝖿𝗅

x

)

.

Hence

𝖼𝗈𝖼𝗈𝗇𝖾

𝒟

⁢

(

E

)

=

E

.

Finally we get an equivalence

(

𝟏

→

E

)

≃

𝖼𝗈𝖼𝗈𝗇𝖾

𝒟

(

E

)

We can now unfold the definitions in order to get the explicit expression of
this map, and we see easily that this is exactly the map we had at the
beginning.

Hence we proved that

𝟏

is a pushout of

𝒟

in

(

n

+

1

)

⁢

-

⁢

𝖳𝗒𝗉𝖾

. Using
uniqueness of pushouts we get that

∥

𝟏

⊔

A

𝟏

∥

n

+

1

=

𝟏

which proves that the suspension of

A

is

(

n

+

1

)

-connected.
∎

Corollary 8.2.2

.

For all

n

:

N

, the sphere

S

n

is

(

n

-

1

)

-connected.

Proof.

We prove this by

induction

on

n

.
For

n

=

0

we have to prove that

𝕊

0

is merely inhabited, which is clear.
Let

n

:

ℕ

be such that

𝕊

n

is

(

n

-

1

)

-connected. By definition

𝕊

n

+

1

is the suspension of

𝕊

n

, hence by the previous lemma

𝕊

n

+

1

is

n

-connected.
∎

Title

8.2 Connectedness of suspensions

\metatable

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