Chapter 8 Notes

Chapter 8 Notes

\toprule

Theorem

Status

\midrule

π

1

⁢

(

𝕊

1

)

\ding

52

\ding

52

π

k

<

n

⁢

(

𝕊

n

)

\ding

52

\ding

52

long-exact-sequence of

homotopy groups

\ding

52

\ding

52

total space

of

Hopf fibration

is

𝕊

3

\ding

52

π

2

⁢

(

𝕊

2

)

\ding

52

\ding

52

π

3

⁢

(

𝕊

2

)

\ding

52

π

n

⁢

(

𝕊

n

)

\ding

52

\ding

52

π

4

⁢

(

𝕊

3

)

\ding

52

Freudenthal suspension theorem

\ding

52

\ding

52

Blakers–Massey theorem

\ding

52

\ding

52

Eilenberg–Mac Lane spaces

K

⁢

(

G

,

n

)

\ding

52

\ding

52

van Kampen theorem

\ding

52

\ding

52

covering spaces

\ding

52

\ding

52

Whitehead’s principle for

n

-types

\ding

52

\ding

52

\bottomrule

Table 1:

Theorems from

homotopy theory

proved by
hand (

\ding

52) and by computer (

\ding

52

\ding

52).

The theorems described in this chapter are standard results in classical
homotopy theory; many are described by

[

hatcher02topology

]

. In these
notes, we review the

development

of the new synthetic proofs of them in

homotopy
type theory

.

\autoref

tab:theorems lists the homotopy-theoretic
theorems that have been proven in homotopy type theory, and whether they
have been computer-checked.
Almost all of these results were developed during the spring term at IAS
in 2013, as part of a significant collaborative effort. Many people
contributed to these results, for example by being the principal author
of a proof, by suggesting problems to work on, by participating in many
discussions and seminars about these problems, or by giving feedback on
results. The following people were the principal authors of the first
homotopy type theory proofs of the above theorems. Unless indicated otherwise, for the
theorems that have been computer-checked, the principal authors were
also the first ones to formalize the proof using a computer proof
assistant.

•

Shulman gave the homotopy-theoretic calculation of

π

1

⁢

(

𝕊

1

)

. Licata later discovered the
encode-decode proof and the encode-decode method.

•

Brunerie calculated

π

k

<

n

⁢

(

𝕊

n

)

.
Licata later gave an encode-decode version.

•

Voevodsky constructed the long exact sequence of homotopy groups.

•

Lumsdaine constructed the Hopf fibration.
Brunerie proved that its total space is

𝕊

3

, thereby calculating

π

2

⁢

(

𝕊

2

)

and

π

3

⁢

(

𝕊

3

)

.

•

Licata and Brunerie gave a direct calculation of

π

n

⁢

(

𝕊

n

)

.

•

Lumsdaine proved the Freudenthal suspension theorem; Licata and
Lumsdaine formalized this proof.

•

Lumsdaine, Finster, and Licata proved the Blakers–Massey theorem;
Lumsdaine, Brunerie, Licata, and Hou formalized it.

•

Licata gave an encode-decode calculation of

π

2

⁢

(

𝕊

2

)

, and a
calculation of

π

n

⁢

(

𝕊

n

)

using the Freudenthal suspension theorem; using

similar

techniques, he constructed

K

⁢

(

G

,

n

)

.

•

Shulman proved the van Kampen theorem; Hou formalized this proof.

•

Licata proved Whitehead’s theorem for

n

-types.

•

Brunerie calculated

π

4

⁢

(

𝕊

3

)

.

•

Hou established the theory of covering spaces and formalized it.

The interplay between homotopy theory and

type theory

was crucial to the
development of these results. For example, the first proof that

π

1

⁢

(

𝕊

1

)

=

ℤ

was the one given in

\autoref

subsec:pi1s1-homotopy-theory, which follows a classical

homotopy

theoretic one. A
type-theoretic

analysis

of this proof resulted in the development of the
encode-decode method. The first calculation of

π

2

⁢

(

𝕊

2

)

also followed
classical methods, but this led quickly to an encode-decode proof of the
result. The encode-decode calculation generalized to

π

n

⁢

(

𝕊

n

)

, which
in turn led to the proof of the Freudenthal suspension theorem, by
combining an encode-decode

argument

with classical homotopy-theoretic
reasoning about

connectedness

, which in turn led to the Blakers–Massey
theorem and Eilenberg–Mac Lane spaces. The rapid development of this
series of results illustrates the promise of our new understanding of
the

connections

between these two subjects.

Title

Chapter 8 Notes

\metatable

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