8.4 Fiber sequences and the long exact sequence

8.4 Fiber sequences and the long exact sequence

If the

codomain

of a function

f

:

X

→

Y

is equipped with a

basepoint

y

0

:

Y

, then we refer to the fiber

F

:

≡

𝖿𝗂𝖻

f

(

y

0

)

of

f

over

y

0

as

the fiber of

f

.
(If

Y

is

connected

, then

F

is determined up to mere equivalence; see

\autoref

ex:unique-fiber.)
We now show that if

X

is also pointed and

f

preserves

basepoints, then there is a

relation

between the

homotopy groups

of

F

,

X

, and

Y

in the form of a

long exact sequence

.
We derive this by way of the

fiber sequence

associated to such an

f

.

Definition 8.4.1

.

A

pointed map

between pointed types

(

X

,

x

0

)

and

(

Y

,

y

0

)

is a
map

f

:

X

→

Y

together with a path

f

0

:

f

⁢

(

x

0

)

=

y

0

.

For any pointed types

(

X

,

x

0

)

and

(

Y

,

y

0

)

, there is a pointed map

(

λ

x

.

y

0

)

:

X

→

Y

which is

constant

at the basepoint.
We call this the

zero map

and sometimes write it as

0

:

X

→

Y

.

Recall that every pointed type

(

X

,

x

0

)

has a

loop space

Ω

⁢

(

X

,

x

0

)

.
We now note that this

operation

is functorial on pointed maps.

Definition 8.4.2

.

Given a pointed map between pointed types

f

:

X

→

Y

, we define a pointed
map

Ω

⁢

f

:

Ω

⁢

X

→

Ω

⁢

Y

by

(

Ω

f

)

(

p

)

:

≡

f

0

-

1

\centerdot

f

(

p

)

\centerdot

f

0

.

The path

(

Ω

⁢

f

)

0

:

(

Ω

⁢

f

)

⁢

(

refl

x

0

)

=

refl

y

0

, which exhibits

Ω

⁢

f

as a pointed map, is the

obvious

path of type

f

0

-

1

\centerdot

f

(

𝗋𝖾𝖿𝗅

x

0

)

\centerdot

f

0

=

𝗋𝖾𝖿𝗅

y

0

.

There is another

functor

on pointed maps, which takes

f

:

X

→

Y

to

𝗉𝗋

1

:

𝖿𝗂𝖻

f

⁢

(

y

0

)

→

X

.
When

f

is pointed, we always consider

𝖿𝗂𝖻

f

⁢

(

y

0

)

to be pointed with basepoint

(

x

0

,

f

0

)

, in which case

𝗉𝗋

1

is also a pointed map, with witness

(

𝗉𝗋

1

)

0

:

≡

𝗋𝖾𝖿𝗅

x

0

.
Thus, this operation can be iterated.

Definition 8.4.3

.

The

fiber sequence

of a pointed map

f

:

X

→

Y

is the

infinite

sequence

of pointed types and pointed maps

\xymatrix

⁢

…

⁢

\ar

⁢

[

r

]

f

(

n

+

1

)

⁢

&

⁢

X

(

n

+

1

)

⁢

\ar

⁢

[

r

]

f

(

n

)

⁢

&

⁢

X

(

n

)

⁢

\ar

-

⁢

f

(

n

-

1

)

⁢

[

r

]

⁢

&

⁢

…

⁢

\ar

⁢

[

r

]

⁢

&

⁢

X

(

2

)

⁢

\ar

-

⁢

f

(

1

)

⁢

[

r

]

⁢

&

⁢

X

(

1

)

⁢

\ar

⁢

[

r

]

f

(

0

)

⁢

&

⁢

X

(

0

)

defined recursively by

X

(

0

)

:

≡

Y

X

(

1

)

:

≡

X

f

(

0

)

:

≡

f

and

X

(

n

+

1

)

:

≡

𝖿𝗂𝖻

f

(

n

-

1

)

(

x

0

(

n

-

1

)

)

f

(

n

)

:

≡

𝗉𝗋

1

:

X

(

n

+

1

)

→

X

(

n

)

.

where

x

0

(

n

)

denotes the basepoint of

X

(

n

)

, chosen recursively as above.

Thus, any adjacent pair of maps in this fiber sequence is of the form

\xymatrix

⁢

X

(

n

+

1

)

≡

𝖿𝗂𝖻

f

(

n

-

1

)

⁢

(

x

0

(

n

-

1

)

)

⁢

\ar

⁢

[

r

⁢

r

]

-

⁢

f

(

n

)

≡

𝗉𝗋

1

⁢

&

⁢

&

⁢

X

(

n

)

⁢

\ar

⁢

[

r

]

f

(

n

-

1

)

⁢

&

⁢

X

(

n

-

1

)

.

In particular, we have

f

(

n

-

1

)

∘

f

(

n

)

=

0

.
We now observe that the types occurring in this sequence are the iterated loop spaces of the base
space

Y

, the

total space

X

, and the fiber

F

:

≡

𝖿𝗂𝖻

f

(

y

0

)

, and similarly for the maps.

Lemma 8.4.4

.

Let

f

:

X

→

Y

be a pointed map of

pointed spaces

. Then:

1.

The fiber of

f

(

1

)

:

≡

𝗉𝗋

1

:

𝖿𝗂𝖻

f

(

y

0

)

→

X

is

equivalent

to

Ω

⁢

Y

.

2.

Similarly, the fiber of

f

(

2

)

:

Ω

⁢

Y

→

𝖿𝗂𝖻

f

⁢

(

y

0

)

is equivalent to

Ω

⁢

X

.

3.

Under these equivalences, the map

f

(

3

)

:

Ω

⁢

X

→

Ω

⁢

Y

is identified with

Ω

⁢

f

∘

(

–

)

-

1

.

Proof.

For

1

, we have

𝖿𝗂𝖻

f

(

1

)

⁢

(

x

0

)

:

≡

∑

z

:

𝖿𝗂𝖻

f

⁢

(

y

0

)

(

𝗉𝗋

1

(

z

)

=

x

0

)

≃

∑

(

x

:

A

)

∑

(

p

:

f

(

x

)

=

y

0

)

(

x

=

x

0

)

(by

\autoref

ex:sigma-assoc)

≃

(

f

(

x

0

)

=

y

0

)

(as

∑

(

x

:

A

)

(

x

=

x

0

)

is

contractible

)

≃

(

y

0

=

y

0

)

(by

(

f

0

⁢

\centerdot

⁢

–

)

)

≡

Ω

⁢

Y

.

Tracing through, we see that this equivalence sends

(

(

x

,

p

)

,

q

)

to

f

0

-

1

\centerdot

f

(

q

-

1

)

\centerdot

p

, while its

inverse

sends

r

:

y

0

=

y

0

to

(

(

x

0

,

f

0

⁢

\centerdot

⁢

r

)

,

𝗋𝖾𝖿𝗅

x

0

)

.
In particular, the basepoint

(

(

x

0

,

f

0

)

,

𝗋𝖾𝖿𝗅

x

0

)

of

𝖿𝗂𝖻

f

(

1

)

⁢

(

x

0

)

is sent to

f

0

-

1

\centerdot

f

(

𝗋𝖾𝖿𝗅

x

0

-

1

)

\centerdot

f

0

, which equals

𝗋𝖾𝖿𝗅

y

0

.
Hence this equivalence is a pointed map (see

\autoref

ex:pointed-equivalences).
Moreover, under this equivalence,

f

(

2

)

is identified with

λ

r

.

(

x

0

,

f

0

\centerdot

r

)

:

Ω

Y

→

𝖿𝗂𝖻

f

(

y

0

)

.

\cref

item:fibseq2 follows immediately by applying

1

to

f

(

1

)

in place of

f

.
Since

(

f

(

1

)

)

0

:

≡

𝗋𝖾𝖿𝗅

x

0

, under this equivalence

f

(

3

)

is identified with the map

Ω

⁢

X

→

𝖿𝗂𝖻

f

(

1

)

⁢

(

x

0

)

defined by

s

↦

(

(

x

0

,

f

0

)

,

s

)

.
Thus, when we compose with the previous equivalence

𝖿𝗂𝖻

f

(

1

)

⁢

(

x

0

)

≃

Ω

⁢

Y

, we see that

s

maps to

f

0

-

1

\centerdot

f

(

s

-

1

)

\centerdot

f

0

, which is by

definition

(

Ω

⁢

f

)

⁢

(

s

-

1

)

, giving

3

.
∎

Thus, the fiber sequence of

f

:

X

→

Y

can be pictured as:

\xymatrix

⁢

@

⁢

C

=

1.5

⁢

p

⁢

c

⁢

…

⁢

\ar

⁢

[

r

]

⁢

&

⁢

Ω

2

⁢

X

⁢

\ar

⁢

[

r

]

-

⁢

Ω

2

⁢

f

⁢

&

⁢

Ω

2

⁢

Y

⁢

\ar

⁢

[

r

]

-

-

Ω

⁢

∂

⁡

&

⁢

Ω

⁢

F

⁢

\ar

⁢

[

r

]

-

-

Ω

⁢

i

⁢

&

⁢

Ω

⁢

X

⁢

\ar

⁢

[

r

]

-

-

Ω

⁢

f

⁢

&

⁢

Ω

⁢

Y

⁢

\ar

⁢

[

r

]

-

⁢

∂

⁡

&

⁢

F

⁢

\ar

⁢

[

r

]

-

⁢

i

⁢

&

⁢

X

⁢

\ar

⁢

[

r

]

f

⁢

&

⁢

Y

.

where the minus signs denote composition with path

inversion

(

–

)

-

1

.
Note that by

\autoref

ex:ap-path-inversion, we have

Ω

⁢

(

Ω

⁢

f

∘

(

–

)

-

1

)

∘

(

–

)

-

1

=

Ω

2

⁢

f

∘

(

–

)

-

1

∘

(

–

)

-

1

=

Ω

2

⁢

f

.

Thus, there are minus signs on the

k

-fold loop maps whenever

k

is odd.

From this fiber sequence we will deduce an

exact sequence

of pointed sets

.
Let

A

and

B

be sets and

f

:

A

→

B

a function, and recall from

\autoref

defn:modal-image the definition of the

image

𝗂𝗆

⁢

(

f

)

, which can be regarded as a subset of

B

:

𝗂𝗆

(

f

)

:

≡

\setof

b

:

B

|

∃

(

a

:

A

)

.

f

(

a

)

=

b

.

If

A

and

B

are moreover pointed with basepoints

a

0

and

b

0

, and

f

is a pointed map, we define the

kernel

of

f

to be the following subset of

A

:

ker

(

f

)

:

≡

\setof

x

:

A

|

f

(

x

)

=

b

0

.

Of course, this is just the fiber of

f

over the basepoint

b

0

; it a subset of

A

because

B

is a set.

Note that any group is a pointed set, with its unit element as basepoint, and any

group homomorphism

is a pointed map.
In this case, the kernel and image agree with the usual notions from

group theory

.

Definition 8.4.5

.

An

exact sequence of pointed sets

is a (possibly

bounded

) sequence of pointed sets and pointed maps:

\xymatrix

⁢

…

⁢

\ar

⁢

[

r

]

⁢

&

⁢

A

(

n

+

1

)

⁢

\ar

⁢

[

r

]

-

⁢

f

(

n

)

⁢

&

⁢

A

(

n

)

⁢

\ar

⁢

[

r

]

f

(

n

-

1

)

⁢

&

⁢

A

(

n

-

1

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

…

such that for every

n

, the image of

f

(

n

)

is equal, as a subset of

A

(

n

)

, to the kernel of

f

(

n

-

1

)

.
In other words, for all

a

:

A

(

n

)

we have

(

f

(

n

-

1

)

(

a

)

=

a

0

(

n

-

1

)

)

⇔

∃

(

b

:

A

(

n

+

1

)

)

.

(

f

(

n

)

(

b

)

=

a

)

.

where

a

0

(

n

)

denotes the basepoint of

A

(

n

)

.

Usually, most or all of the pointed sets in an exact sequence are groups, and often

abelian groups

.
When we speak of an

exact sequence of groups

, it is assumed moreover that the maps are group homomorphisms and not just pointed maps.

Theorem 8.4.6

.

Let

f

:

X

→

Y

be a pointed map between pointed spaces with fiber

F

:

≡

fib

f

(

y

0

)

.
Then we have the following long exact sequence, which consists of groups except for the last three terms, and abelian groups except for the last six.

\xymatrix

⁢

@

⁢

R

=

1.2

⁢

p

⁢

c

⁢

@

⁢

C

=

3

⁢

p

⁢

c

⁢

⋮

⁢

&

⁢

⋮

⁢

&

⁢

⋮

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

π

k

⁢

(

F

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

k

⁢

(

X

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

k

⁢

(

Y

)

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

⋮

⁢

&

⁢

⋮

⁢

&

⁢

⋮

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

π

2

⁢

(

F

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

2

⁢

(

X

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

2

⁢

(

Y

)

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

π

1

⁢

(

F

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

1

⁢

(

X

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

1

⁢

(

Y

)

⁢

\ar

⁢

[

l

⁢

l

⁢

d

]

⁢

π

0

⁢

(

F

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

0

⁢

(

X

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

0

⁢

(

Y

)

Proof.

We begin by showing that the 0-truncation of a fiber sequence is an exact sequence of pointed sets.
Thus, we need to show that for any adjacent pair of maps in a fiber sequence:

\xymatrix

⁢

𝖿𝗂𝖻

f

⁢

(

z

0

)

⁢

\ar

-

⁢

g

⁢

[

r

]

⁢

&

⁢

W

⁢

\ar

-

⁢

f

⁢

[

r

]

⁢

&

⁢

Z

with

g

:

≡

𝗉𝗋

1

, the sequence

\xymatrix

∥

𝖿𝗂𝖻

f

(

z

0

)

∥

0

\ar

-

∥

g

∥

0

[

r

]

&

∥

W

∥

0

\ar

-

∥

f

∥

0

[

r

]

&

∥

Z

∥

0

is exact, i.e. that

𝗂𝗆

(

∥

g

∥

0

)

⊆

ker

(

∥

f

∥

0

)

and

ker

(

∥

f

∥

0

)

⊆

𝗂𝗆

(

∥

g

∥

0

)

.

The first

inclusion

is equivalent to

∥

g

∥

0

∘

∥

f

∥

0

=

0

, which holds by functoriality of

∥

–

∥

0

and the fact that

g

∘

f

=

0

.
For the second, we assume

w

′

:

∥

W

∥

0

and

p

′

:

∥

f

∥

0

(

w

′

)

=

|

z

0

|

0

and show there merely exists

t

:

𝖿𝗂𝖻

f

⁢

(

z

0

)

such that

g

⁢

(

t

)

=

w

′

.
Since our goal is a mere proposition, we can assume that

w

′

is of the
form

|

w

|

0

for some

w

:

W

.
Now by

\autoref

thm:path-truncation,

p

′

:

|

f

(

w

)

|

0

=

|

z

0

|

0

yields

p

′′

:

∥

f

(

w

)

=

z

0

∥

-

1

, so by a further

truncation

induction

we may assume some

p

:

f

⁢

(

w

)

=

z

0

.
But now we have

|

(

w

,

p

)

|

0

:

|

𝖿𝗂𝖻

f

(

z

0

)

|

0

whose image under

∥

g

∥

0

is

|

w

|

0

≡

w

′

, as desired.

Thus, applying

∥

–

∥

0

to the fiber sequence of

f

, we obtain a long exact sequence involving the pointed sets

π

k

⁢

(

F

)

,

π

k

⁢

(

X

)

, and

π

k

⁢

(

Y

)

in the desired order.
And of course,

π

k

is a group for

k

≥

1

, being the 0-truncation of a loop space, and an abelian group for

k

≥

2

by the Eckmann–Hilton

argument

(

\autoref

thm:EckmannHilton).
Moreover,

\autoref

thm:fiber-of-the-fiber allows us to identify the maps

π

k

⁢

(

F

)

→

π

k

⁢

(

X

)

and

π

k

⁢

(

X

)

→

π

k

⁢

(

Y

)

in this exact sequence as

(

-

1

)

k

⁢

π

k

⁢

(

i

)

and

(

-

1

)

k

⁢

π

k

⁢

(

f

)

respectively.

More generally, every map in this long exact sequence except the last three is of the form

∥

Ω

⁢

h

∥

0

or

∥

-

Ω

⁢

h

∥

0

for some

h

.
In the former case it is a group homomorphism, while in the latter case it is a

homomorphism

if the groups are

abelian

; otherwise it is an “

anti-homomorphism

”.
However, the kernel and image of a group homomorphism are unchanged when we replace it by its

negative

, and hence so is the exactness of any sequence involving it.
Thus, we can modify our long exact sequence to obtain one involving

π

k

⁢

(

i

)

and

π

k

⁢

(

f

)

directly and in which all the maps are group homomorphisms (except the last three).
∎

The usual

properties

of exact sequences of abelian groups can be proved as
usual. In particular we have:

Lemma 8.4.7

.

Suppose given an exact sequence of abelian groups:

\xymatrix

⁢

K

⁢

\ar

⁢

[

r

]

⁢

&

⁢

G

⁢

\ar

f

⁢

[

r

]

⁢

&

⁢

H

⁢

\ar

⁢

[

r

]

⁢

&

⁢

Q

.

1.

If

K

=

0

, then

f

is

injective

.

2.

If

Q

=

0

, then

f

is

surjective

.

3.

If

K

=

Q

=

0

, then

f

is an

isomorphism

.

Proof.

Since the kernel of

f

is the image of

K

→

G

, if

K

=

0

then the kernel of

f

is

{

0

}

;
hence

f

is injective because it’s a group

morphism

.
Similarly, since the image of

f

is the kernel of

H

→

Q

, if

Q

=

0

then the image of

f

is all of

H

, so

f

is surjective.
Finally,

3

follows from

1

and

2

by

\autoref

thm:mono-surj-equiv.
∎

As an immediate application, we can now quantify in what way

n

-connectedness of a map is

stronger

than inducing an equivalence on

n

-truncations.

Corollary 8.4.8

.

Let

f

:

A

→

B

be

n

-connected and

a

:

A

, and define

b

:

≡

f

(

a

)

. Then:

1.

If

k

≤

n

, then

π

k

⁢

(

f

)

:

π

k

⁢

(

A

,

a

)

→

π

k

⁢

(

B

,

b

)

is an isomorphism.

2.

If

k

=

n

+

1

, then

π

k

⁢

(

f

)

:

π

k

⁢

(

A

,

a

)

→

π

k

⁢

(

B

,

b

)

is surjective.

Proof.

As part of the long exact sequence, for each

k

we have an exact sequence

\xymatrix

⁢

π

k

⁢

(

𝖿𝗂𝖻

f

⁢

(

b

)

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

k

⁢

(

A

,

a

)

⁢

\ar

f

⁢

[

r

]

⁢

&

⁢

π

k

⁢

(

B

,

b

)

⁢

\ar

⁢

[

r

]

⁢

&

⁢

π

k

-

1

⁢

(

𝖿𝗂𝖻

f

⁢

(

b

)

)

.

Now since

f

is

n

-connected,

∥

𝖿𝗂𝖻

f

⁢

(

b

)

∥

n

is contractible.
Therefore, if

k

≤

n

, then

π

k

(

𝖿𝗂𝖻

f

(

b

)

)

=

∥

Ω

k

(

𝖿𝗂𝖻

f

(

b

)

)

∥

0

=

Ω

k

(

∥

𝖿𝗂𝖻

f

(

b

)

∥

k

)

is also contractible.
Thus,

π

k

⁢

(

f

)

is an isomorphism for

k

≤

n

by

\autoref

thm:ses

3

, while for

k

=

n

+

1

it is surjective by

\autoref

thm:ses

2

.
∎

In

\autoref

sec:whitehead we will see that the

converse

of

\autoref

thm:conn-pik also holds.

Title

8.4 Fiber sequences and the long exact sequence

\metatable

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