aarjaneiro/planetmath huggingface dataset:
3.9 The principle of unique choice

3.9 The principle of unique choice

The following observation is

trivial

, but very useful.

Lemma 3.9.1

.

If

P

is a mere proposition, then

P

≃

∥

P

∥

.

Proof.

Of course, we have

P

→

∥

P

∥

by definition.
And since

P

is a mere proposition, the

universal property

of

∥

P

∥

applied to

𝗂𝖽

P

:

P

→

P

yields

∥

P

∥

→

P

.
These

functions

are

quasi-inverses

by Lemma 3.3.3 (

http://planetmath.org/33merepropositions#Thmprelem2

).
∎

Among its important

consequences

is the following.

Corollary 3.9.2

(The principle of unique choice).

Suppose a type family

P

:

A

→

U

such that

1.

For each

x

, the type

P

⁢

(

x

)

is a mere proposition, and

2.

For each

x

we have

∥

P

⁢

(

x

)

∥

.

Then we have

∏

(

x

:

A

)

P

⁢

(

x

)

.

Proof.

Immediate from the two

assumptions

and the previous lemma.
∎

The corollary also encapsulates a very useful technique of reasoning.
Namely, suppose we know that

∥

A

∥

, and we want to use this to

construct

an element of some other type

B

.
We would like to use an element of

A

in our construction of an element of

B

, but this is allowed only if

B

is a mere proposition, so that we can apply the

induction

principle for the propositional truncation

∥

A

∥

; the most we could hope to do in general is to show

∥

B

∥

.
Instead, we can extend

B

with additional data which characterizes

uniquely

the object we wish to construct.
Specifically, we define a

predicate

Q

:

B

→

𝒰

such that

∑

(

x

:

B

)

Q

⁢

(

x

)

is a mere proposition.
Then from an element of

A

we construct an

element

b

:

B

such that

Q

⁢

(

b

)

, hence from

∥

A

∥

we can construct

∥

∑

(

x

:

B

)

Q

⁢

(

x

)

∥

, and because

∥

∑

(

x

:

B

)

Q

⁢

(

x

)

∥

is

equivalent

to

∑

(

x

:

B

)

Q

⁢

(

x

)

an element of

B

may be projected from it.
An example can be found in

http://planetmath.org/node/87853

Exercise 3.19.

A similar issue arises in set-theoretic mathematics, although it manifests slightly
differently. If we are trying to define a function

f

:

A

→

B

, and depending on an
element

a

:

A

we are able to prove mere

existence

of some

b

:

B

, we are not done yet
because we need to actually pinpoint an element of

B

, not just prove its existence.
One option is of course to refine the

argument

to unique existence of

b

:

B

, as we did in

type theory

. But in

set theory

the problem can often be avoided more simply by an application of the

axiom of choice

, which picks the required elements for us.
In

homotopy type theory

, however, quite apart from any desire to avoid choice, the available forms of choice are simply less applicable, since they require that the domain of choice be a

set

.
Thus, if

A

is not a set (such as perhaps a

universe

𝒰

), there is no

consistent

form of choice that will allow us to simply pick an element of

B

for each

a

:

A

to use in defining

f

⁢

(

a

)

.

Title

3.9 The principle of unique choice

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4.3 Bi-invertible maps

4.3 Bi-invertible maps

Using the language introduced in §4.2 (

http://planetmath.org/42halfadjointequivalences

), we can restate the

definition

proposed in §2.4 (

http://planetmath.org/24homotopiesandequivalences

) as follows.

Definition 4.3.1

.

We say

f

:

A

→

B

is

bi-invertible

if it has both a

left inverse

and a

right inverse

:

𝖻𝗂𝗂𝗇𝗏

(

f

)

:

≡

𝗅𝗂𝗇𝗏

(

f

)

×

𝗋𝗂𝗇𝗏

(

f

)

.

In §2.4 (

http://planetmath.org/24homotopiesandequivalences

) we proved that

𝗊𝗂𝗇𝗏

⁢

(

f

)

→

𝖻𝗂𝗂𝗇𝗏

⁢

(

f

)

and

𝖻𝗂𝗂𝗇𝗏

⁢

(

f

)

→

𝗊𝗂𝗇𝗏

⁢

(

f

)

.
What remains is the following.

Theorem 4.3.2

.

For any

f

:

A

→

B

, the type

biinv

⁢

(

f

)

is a mere proposition.

Proof.

We may suppose

f

to be bi-invertible and show that

𝖻𝗂𝗂𝗇𝗏

⁢

(

f

)

is

contractible

.
But since

𝖻𝗂𝗂𝗇𝗏

⁢

(

f

)

→

𝗊𝗂𝗇𝗏

⁢

(

f

)

, by Lemma 4.2.9 (

http://planetmath.org/42halfadjointequivalences#Thmprelem4

) in this case both

𝗅𝗂𝗇𝗏

⁢

(

f

)

and

𝗋𝗂𝗇𝗏

⁢

(

f

)

are contractible, and the product of contractible types is contractible.
∎

Note that this also fits the proposal made at the beginning of §4.2 (

http://planetmath.org/42halfadjointequivalences

): we combine

g

and

η

into a contractible type and add an additional datum which combines with

ϵ

into a contractible type.
The

difference

is that instead of adding a

higher

datum (a 2-dimensional path) to combine with

ϵ

, we add a

lower

one (a right inverse that is separate from the left inverse).

Corollary 4.3.3

.

For any

f

:

A

→

B

we have

biinv

⁢

(

f

)

≃

ishae

⁢

(

f

)

.

Proof.

We have

𝖻𝗂𝗂𝗇𝗏

⁢

(

f

)

→

𝗊𝗂𝗇𝗏

⁢

(

f

)

→

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

and

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

→

𝗊𝗂𝗇𝗏

⁢

(

f

)

→

𝖻𝗂𝗂𝗇𝗏

⁢

(

f

)

.
Since both

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

and

𝖻𝗂𝗂𝗇𝗏

⁢

(

f

)

are mere propositions, the

equivalence

follows from Lemma 3.3.3 (

http://planetmath.org/33merepropositions#Thmprelem2

).
∎

Title

4.3 Bi-invertible maps

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4.5 On the definition of equivalences

4.5 On the definition of equivalences

We have shown that all three

definitions

of equivalence

satisfy

the three desirable

properties

and are pairwise

equivalent

:

𝗂𝗌𝖢𝗈𝗇𝗍𝗋

⁢

(

f

)

≃

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

≃

𝖻𝗂𝗂𝗇𝗏

⁢

(

f

)

.

(There are yet more possible definitions of equivalence, but we will stop with these three.
See

http://planetmath.org/node/87824

Exercise 3.11 and the exercises in this chapter for some more.)
Thus, we may choose any one of them as “the” definition of

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

f

)

.
For definiteness, we choose to define

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

(

f

)

:

≡

𝗂𝗌𝗁𝖺𝖾

(

f

)

.

This choice is advantageous for formalization, since

𝗂𝗌𝗁𝖺𝖾

⁢

(

f

)

contains the most directly useful data.
On the other hand, for other purposes,

𝖻𝗂𝗂𝗇𝗏

⁢

(

f

)

is often easier to deal with, since it contains no 2-dimensional paths and its two symmetrical halves can be treated independently.
However, for purposes of this book, the specific choice will make little

difference

.

In the rest of this chapter, we study some other properties and

characterizations

of equivalences.

Title

4.5 On the definition of equivalences

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4. Equivalences

4. Equivalences

We now study in more detail the notion of

equivalence of types

that was introduced briefly in §2.4 (

http://planetmath.org/24homotopiesandequivalences

).
Specifically, we will give several different ways to define a type

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

f

)

having the properties mentioned there.
Recall that we wanted

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

f

)

to have the following properties, which we restate here:

1.

𝗊𝗂𝗇𝗏

⁢

(

f

)

→

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

f

)

.

2.

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

f

)

→

𝗊𝗂𝗇𝗏

⁢

(

f

)

.

3.

𝗂𝗌𝖾𝗊𝗎𝗂𝗏

⁢

(

f

)

is a mere proposition.

Here

𝗊𝗂𝗇𝗏

⁢

(

f

)

denotes the type of

quasi-inverses

to

f

:

∑

g

:

B

→

A

(

(

f

∘

g

∼

𝗂𝖽

B

)

×

(

g

∘

f

∼

𝗂𝖽

A

)

)

.

By function extensionality, it follows that

𝗊𝗂𝗇𝗏

⁢

(

f

)

is

equivalent

to the type

∑

g

:

B

→

A

(

(

f

∘

g

=

𝗂𝖽

B

)

×

(

g

∘

f

=

𝗂𝖽

A

)

)

.

We will define three different types having properties

1

–

3

, which we call

•

half adjoint equivalences,

•

bi-invertible maps,

and

•

contractible

functions

.

We will also show that all these types are equivalent.
These names are intentionally somewhat cumbersome, because after we know that they are all equivalent and have properties

1

–

3

, we will revert to saying simply “equivalence” without needing to specify which particular definition we choose.
But for purposes of the comparisons in this chapter, we need different names for each definition.

Before we examine the different notions of equivalence, however, we give a little more explanation of why a different

concept

than quasi-invertibility is needed.

Title

4. Equivalences

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5. Induction

5. Induction

In

http://planetmath.org/node/87533

Chapter 1, we introduced many ways to form new types from old ones.
Except for (dependent) function types and

universes

, all these rules are special cases of the general notion of

inductive definition

.
In this chapter we study inductive definitions more generally.

Title

5.

Induction

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6.3 The interval

6.3 The interval

The

interval

, which we denote

I

, is perhaps an even simpler

higher inductive type

than the circle.
It is generated by:

•

a point

0

I

:

I

,

•

a point

1

I

:

I

, and

•

a path

𝗌𝖾𝗀

:

0

I

=

I

1

I

.

The

recursion

principle for the interval says that given a type

B

along with

•

a point

b

0

:

B

,

•

a point

b

1

:

B

, and

•

a path

s

:

b

0

=

b

1

,

there is a

function

f

:

I

→

B

such that

f

⁢

(

0

I

)

≡

b

0

,

f

⁢

(

1

I

)

≡

b

1

, and

f

(

𝗌𝖾𝗀

)

=

s

.
The

induction

principle says that given

P

:

I

→

𝒰

along with

•

a point

b

0

:

P

⁢

(

0

I

)

,

•

a point

b

1

:

P

⁢

(

1

I

)

, and

•

a path

s

:

b

0

=

𝗌𝖾𝗀

P

b

1

,

there is a function

f

:

∏

(

x

:

I

)

P

⁢

(

x

)

such that

f

⁢

(

0

I

)

≡

b

0

,

f

⁢

(

1

I

)

≡

b

1

, and

𝖺𝗉𝖽

f

(

𝗌𝖾𝗀

)

=

s

.

Regarded purely up to

homotopy

, the interval is not really interesting:

Lemma 6.3.1

.

The type

I

is

contractible

.

Proof.

We prove that for all

x

:

I

we have

x

=

I

1

I

. In other words we want a
function

f

of type

∏

(

x

:

I

)

(

x

=

I

1

I

)

. We begin to define

f

in the following way:

f

⁢

(

0

I

)

:

≡

𝗌𝖾𝗀

:

0

I

=

I

1

I

,

f

⁢

(

1

I

)

:

≡

𝗋𝖾𝖿𝗅

1

I

:

1

I

=

I

1

I

.

It remains to define

𝖺𝗉𝖽

f

(

𝗌𝖾𝗀

)

, which must have type

𝗌𝖾𝗀

=

𝗌𝖾𝗀

λ

⁢

x

.

x

=

I

1

I

𝗋𝖾𝖿𝗅

1

I

.
By

definition

this type is

𝗌𝖾𝗀

*

(

𝗌𝖾𝗀

)

=

1

I

=

I

1

I

𝗋𝖾𝖿𝗅

1

I

, which in turn is

equivalent

to

𝗌𝖾𝗀

-

1

⁢

\centerdot

⁢

𝗌𝖾𝗀

=

𝗋𝖾𝖿𝗅

1

I

.
But there is a

canonical

element of that type, namely the proof that path

inverses

are in fact inverses.
∎

However, type-theoretically the interval does still have some interesting features, just like the topological interval in classical

homotopy theory

.
For

instance

, it enables us to give an easy proof of function extensionality.
(Of course, as in §4.9 (

http://planetmath.org/49univalenceimpliesfunctionextensionality

), for the duration of the following proof we suspend our overall

assumption

of the function extensionality axiom.)

Lemma 6.3.2

.

If

f

,

g

:

A

→

B

are two functions such that

f

⁢

(

x

)

=

g

⁢

(

x

)

for every

x

:

A

, then

f

=

g

in the type

A

→

B

.

Proof.

Let’s call the proof we have

p

:

∏

(

x

:

A

)

(

f

(

x

)

=

g

(

x

)

)

. For all

x

:

A

we define
a function

p

~

x

:

I

→

B

by

p

~

x

⁢

(

0

I

)

:

≡

f

(

x

)

,

p

~

x

⁢

(

1

I

)

:

≡

g

(

x

)

,

p

~

x

(

𝗌𝖾𝗀

)

:=

p

⁢

(

x

)

.

We now define

q

:

I

→

(

A

→

B

)

by

q

(

i

)

:

≡

(

λ

x

.

p

~

x

(

i

)

)

Then

q

⁢

(

0

I

)

is the function

λ

⁢

x

.

p

~

x

⁢

(

0

I

)

, which is equal to

f

because

p

~

x

⁢

(

0

I

)

is defined by

f

⁢

(

x

)

.
Similarly, we have

q

⁢

(

1

I

)

=

g

, and hence

q

(

𝗌𝖾𝗀

)

:

f

=

(

A

→

B

)

g

∎

Title

6.3 The interval

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7. Homotopy n-types

7. Homotopy n-types

One of the basic notions of

homotopy theory

is that of a

homotopy

n

-type

: a space containing no interesting homotopy above

dimension

n

.
For

instance

, a homotopy

0

-type is essentially a set, containing no nontrivial paths, while a homotopy

1

-type may contain nontrivial paths, but no nontrivial paths between paths.
Homotopy

n

-types are also called

n

-truncated spaces

.
We have mentioned this notion already in §3.1 (

http://planetmath.org/31setsandntypes

); our first goal in this chapter is to give it a precise definition in

homotopy type theory

.

A dual notion to truncatedness is

connectedness

: a space is

n

-connected

if it has no interesting homotopy in dimensions

n

and

below

.
For instance, a space is

0

-connected (also called just “connected”) if it has only one

connected component

, and

1

-connected (also called “

simply connected

”) if it also has no nontrivial loops (though it may have nontrivial higher loops between loops).

The duality between truncatedness and connectedness is most easily seen by extending both notions to maps.
We call a map

n

-truncated

or

n

-connected

if all its fibers are so.
Then

n

-connected and

n

-truncated maps form the two classes of maps in an

orthogonal factorization system

,

i.e. every map factors uniquely as an

n

-connected map followed by an

n

-truncated one.

In the case

n

=

-

1

, the

n

-truncated maps are the

embeddings

and the

n

-connected maps are the

surjections

, as defined in §4.6 (

http://planetmath.org/46surjectionsandembeddings

).
Thus, the

n

-connected

factorization system

is a massive

generalization

of the standard image

factorization

of a

function

between sets into a surjection followed by an injection.
At the end of this chapter, we sketch briefly an even more general theory: any type-theoretic

modality

gives rise to an analogous factorization system.

Title

7. Homotopy n-types

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8.10 Additional Results

8.10 Additional Results

Though we do not present the proofs in this chapter, following results have also been established in

homotopy type theory

.

Theorem 8.10.1

.

There exists a

k

such that for all

n

≥

3

,

π

n

+

1

⁢

(

S

n

)

=

Z

k

.

Notes on the proof..

The proof consists of a calculation of

π

4

⁢

(

𝕊

3

)

, together with an
appeal to stability (

\cref

cor:stability-spheres). In the classical
statement of this result,

k

is

2

. While we have not yet checked that

k

is in fact

2

, our calcluation of

π

4

⁢

(

𝕊

3

)

is constructive,
like all the rest of the proofs in this chapter.
(More precisely, it doesn’t use any additional axioms such as

𝖫𝖤𝖬

or

𝖠𝖢

, making it as constructive as
univalence and

higher inductive types

are.) Thus, given a
computational

interpretation

of homotopy type theory, we could run the
proof on a computer to verify that

k

is

2

. This example is quite
intriguing, because it is the first calculation of a

homotopy group

for which we have not needed to know the answer in advance.
∎

Theorem 8.10.2

(Blakers–Massey theorem).

Suppose we are given maps

f

:

C

→

X

, and

g

:

C

→

Y

. Taking first the pushout

X

⊔

C

Y

of

f

and

g

and then the

pullback

of its

inclusions

inl

:

X

→

X

⊔

C

Y

←

Y

:

inr

, we have an

induced map

C

→

X

×

(

X

⊔

C

Y

)

Y

.

If

f

is

i

-

connected

and

g

is

j

-connected, then this induced map is

(

i

+

j

)

-connected. In other words, for any points

x

:

X

,

y

:

Y

, the corresponding fiber

C

x

,

y

of

(

f

,

g

)

:

C

→

X

×

Y

gives an approximation to the

path space

inl

⁢

(

x

)

=

X

⊔

C

Y

inr

⁢

(

y

)

in the pushout.

It should be noted that in classical

algebraic topology

, the Blakers–Massey

theorem

is often stated in a somewhat different form, where the maps

f

and

g

are replaced by inclusions of subcomplexes of

CW complexes

, and the

homotopy

pushout and homotopy pullback by a union and

intersection

, respectively.
In order to express the theorem in homotopy type theory, we have to replace notions of this sort with ones that are homotopy-invariant.
We have seen another example of this in the

van Kampen theorem

(

\autoref

sec:van-kampen), where we had to replace a union of

open subsets

by a homotopy pushout.

Theorem 8.10.3

(Eilenberg–Mac Lane Spaces).

For any

abelian group

G

and positive integer

n

, there is an

n

-type

K

⁢

(

G

,

n

)

such that

π

n

⁢

(

K

⁢

(

G

,

n

)

)

=

G

, and

π

k

⁢

(

K

⁢

(

G

,

n

)

)

=

0

for

k

≠

n

.

Theorem 8.10.4

(Covering spaces).

For a connected space

A

, there is an

equivalence

between covering spaces over

A

and sets with an action of

π

1

⁢

(

A

)

.

Title

8.10 Additional Results

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8.1.1 Getting started

8.1.1 Getting started

It is not too hard to define

functions

in both directions between

Ω

⁢

(

𝕊

1

)

and

ℤ

.
By specializing

\autoref

thm:looptothe to

𝗅𝗈𝗈𝗉

:

𝖻𝖺𝗌𝖾

=

𝖻𝖺𝗌𝖾

, we have a function

𝗅𝗈𝗈𝗉

–

:

ℤ

→

(

𝖻𝖺𝗌𝖾

=

𝖻𝖺𝗌𝖾

)

defined (loosely speaking) by

𝗅𝗈𝗈𝗉

n

=

{

𝗅𝗈𝗈𝗉

⁢

\centerdot

⁢

𝗅𝗈𝗈𝗉

⁢

\centerdot

⁢

⋯

⁢

\centerdot

⁢

𝗅𝗈𝗈𝗉

⏟

n

if

n

>

0

,

𝗅𝗈𝗈𝗉

-

1

⁢

\centerdot

⁢

𝗅𝗈𝗈𝗉

-

1

⁢

\centerdot

⁢

⋯

⁢

\centerdot

⁢

𝗅𝗈𝗈𝗉

-

1

⏟

-

n

if

n

<

0

,

𝗋𝖾𝖿𝗅

𝖻𝖺𝗌𝖾

if

n

=

0

.

Defining a function

g

:

Ω

⁢

(

𝕊

1

)

→

ℤ

in the other direction is a bit trickier.
Note that the successor function

𝗌𝗎𝖼𝖼

:

ℤ

→

ℤ

is an

equivalence

,
and hence induces a path

𝗎𝖺

⁢

(

𝗌𝗎𝖼𝖼

)

:

ℤ

=

ℤ

in the

universe

𝒰

.
Thus, the

recursion

principle of

𝕊

1

induces a map

c

:

𝕊

1

→

𝒰

by

c

(

𝖻𝖺𝗌𝖾

)

:

≡

ℤ

and

𝖺𝗉

c

⁢

(

𝗅𝗈𝗈𝗉

)

:=

𝗎𝖺

⁢

(

𝗌𝗎𝖼𝖼

)

.
Then we have

𝖺𝗉

c

:

(

𝖻𝖺𝗌𝖾

=

𝖻𝖺𝗌𝖾

)

→

(

ℤ

=

ℤ

)

, and we can define

g

(

p

)

:

≡

𝗍𝗋𝖺𝗇𝗌𝗉𝗈𝗋𝗍

X

↦

X

(

𝖺𝗉

c

(

p

)

,

0

)

.

With these

definitions

, we can even prove that

g

⁢

(

𝗅𝗈𝗈𝗉

n

)

=

n

for any

n

:

ℤ

, using the

induction

principle

\autoref

thm:sign-induction for

n

.
(We will prove something more general a little later on.)
However, the other equality

𝗅𝗈𝗈𝗉

g

⁢

(

p

)

=

p

is significantly harder.
The

obvious

thing to try is path induction, but path induction does not apply to loops such as

p

:

(

𝖻𝖺𝗌𝖾

=

𝖻𝖺𝗌𝖾

)

that have

both

endpoints

fixed!
A new idea is required, one which can be explained both in terms of classical

homotopy theory

and in terms of

type theory

.
We begin with the former.

Title

8.1.1 Getting started

\metatable

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8.1 π_1⁢(S_1)

8.1

π

1

⁢

(

S

1

)

In this

section

, our goal is to show that

π

1

⁢

(

𝕊

1

)

=

ℤ

.
In fact, we will show that the

loop space

Ω

⁢

(

𝕊

1

)

is

equivalent

to

ℤ

.
This is a

stronger

statement, because

π

1

(

𝕊

1

)

=

∥

Ω

(

𝕊

1

)

∥

0

by
definition; so if

Ω

⁢

(

𝕊

1

)

=

ℤ

, then

∥

Ω

(

𝕊

1

)

∥

0

=

∥

ℤ

∥

0

by

congruence

, and

ℤ

is a set by definition (being a set-quotient; see

\autoref

defn-Z,

\autoref

Z-quotient-by-canonical-representatives), so

∥

ℤ

∥

0

=

ℤ

.
Moreover, knowing that

Ω

⁢

(

𝕊

1

)

is a set will imply that

π

n

⁢

(

𝕊

1

)

is

trivial

for

n

>

1

, so we will actually have calculated

all

the

homotopy groups

of

𝕊

1

.

Title

8.1

π

1

⁢

(

S

1

)

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8.3 π_{k≤n} of an n-connected space and π_{k<n}⁢(S^n)

8.3

π

k

≤

n

of an n-connected space and

π

k

<

n

⁢

(

S

n

)

Let

(

A

,

a

)

be a pointed type and

n

:

ℕ

. Recall from

\autoref

thm:homotopy-groups that if

n

>

0

the set

π

n

⁢

(

A

,

a

)

has a group

structure

, and if

n

>

1

the group is abelian.

We can now say something about

homotopy groups

of

n

-truncated and

n

-

connected

types.

Lemma 8.3.1

.

If

A

is

n

-truncated and

a

:

A

, then

π

k

⁢

(

A

,

a

)

=

1

for all

k

>

n

.

Proof.

The

loop space

of an

n

-type is an

(

n

-

1

)

-type, hence

Ω

k

⁢

(

A

,

a

)

is an

(

n

-

k

)

-type, and we have

(

n

-

k

)

≤

-

1

so

Ω

k

⁢

(

A

,

a

)

is a mere proposition. But

Ω

k

⁢

(

A

,

a

)

is inhabited,
so it is actually

contractible

and

π

k

(

A

,

a

)

=

∥

Ω

k

(

A

,

a

)

∥

0

=

∥

𝟏

∥

0

=

𝟏

.
∎

Lemma 8.3.2

.

If

A

is

n

-connected and

a

:

A

, then

π

k

⁢

(

A

,

a

)

=

1

for all

k

≤

n

.

Proof.

We have the following

sequence

of equalities:

π

k

(

A

,

a

)

=

∥

Ω

k

(

A

,

a

)

∥

0

=

Ω

k

(

∥

(

A

,

a

)

∥

k

)

=

Ω

k

(

∥

∥

(

A

,

a

)

∥

n

∥

k

)

=

Ω

k

(

∥

𝟏

∥

k

)

=

Ω

k

(

𝟏

)

=

𝟏

.

The third equality uses the fact that

k

≤

n

in order to use that

∥

–

∥

k

∘

∥

–

∥

n

=

∥

–

∥

k

and the fourth equality uses the fact that

A

is

n

-connected.
∎

Corollary 8.3.3

.

π

k

⁢

(

𝕊

n

)

=

𝟏

for

k

<

n

.

Proof.

The sphere

𝕊

n

is

(

n

-

1

)

-connected by

\autoref

cor:sn-connected, so
we can apply

\autoref

lem:pik-nconnected.
∎

Title

8.3

π

k

≤

n

of an n-connected space and

π

k

<

n

⁢

(

S

n

)

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8.5 The Hopf fibration

8.5 The Hopf fibration

In this

section

we will define the

Hopf fibration

.

Theorem 8.5.1

(Hopf Fibration).

There is a

fibration

H

over

S

2

whose fiber over the

basepoint

is

S

1

and
whose total space is

S

3

.

The Hopf fibration will allow us to compute several homotopy groups of
spheres.
Indeed, it yields the following long exact sequence

of homotopy groups
(see

\autoref

sec:long-exact-sequence-homotopy-groups):

\xymatrix

⁢

@

⁢

R

=

1.2

⁢

p

⁢

c

⁢

π

k

⁢

(

𝕊

1

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

k

⁢

(

𝕊

3

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

k

⁢

(

𝕊

2

)

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

⋮

⁢

&

⁢

⋮

⁢

&

⁢

⋮

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

π

2

⁢

(

𝕊

1

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

2

⁢

(

𝕊

3

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

2

⁢

(

𝕊

2

)

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

π

1

⁢

(

𝕊

1

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

1

⁢

(

𝕊

3

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

1

⁢

(

𝕊

2

)

We’ve already computed all

π

n

⁢

(

𝕊

1

)

, and

π

k

⁢

(

𝕊

n

)

for

k

<

n

, so this
becomes the following:

\xymatrix

⁢

@

⁢

R

=

1.2

⁢

p

⁢

c

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

k

⁢

(

𝕊

3

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

k

⁢

(

𝕊

2

)

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

⋮

⁢

&

⁢

⋮

⁢

&

⁢

⋮

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

3

⁢

(

𝕊

3

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

3

⁢

(

𝕊

2

)

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

2

⁢

(

𝕊

2

)

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

ℤ

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

⁢

\ar

⁢

[

r

]

⁢

&

⁢

0

In particular we get the following result:

Corollary 8.5.2

.

We have

π

2

⁢

(

S

2

)

≃

Z

and

π

k

⁢

(

S

3

)

≃

π

k

⁢

(

S

2

)

for
every

k

≥

3

(where the map is induced by the Hopf fibration, seen as a map
from the total space

S

3

to the

base space

S

2

).

In fact, we can say more: the fiber sequence of the Hopf fibration will show that

Ω

3

⁢

(

𝕊

3

)

is the fiber of a map from

Ω

3

⁢

(

𝕊

2

)

to

Ω

2

⁢

(

𝕊

1

)

.
Since

Ω

2

⁢

(

𝕊

1

)

is

contractible

, we have

Ω

3

⁢

(

𝕊

3

)

≃

Ω

3

⁢

(

𝕊

2

)

.
In classical

homotopy theory

, this fact would be a

consequence

of

\autoref

cor:pis2-hopf and Whitehead’s

theorem

, but Whitehead’s theorem is not necessarily valid in

homotopy type theory

(see

\autoref

sec:whitehead).
We will not use the more precise version here though.

Title

8.5 The Hopf fibration

\metatable

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