Puppe sequence

In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration). Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.

Exact Puppe sequence

A sequence of pointed spaces and pointed maps

⋯

→

X

n

+

1

→

X

n

→

X

n

−

1

→

…

{\displaystyle \dots \to X_{n+1}\to X_{n}\to X_{n-1}\to \dots }

is called exact if the induced sequence

⋯

→

[

Z

,

X

n

+

1

]

→

[

Z

,

X

n

]

→

[

Z

,

X

n

−

1

]

→

…

{\displaystyle \dots \to [Z,X_{n+1}]\to [Z,X_{n}]\to [Z,X_{n-1}]\to \dots }

is exact as a sequence of pointed sets (taking the kernel of a map to be those elements mapped to the basepoint) for every pointed space

Z

{\displaystyle Z}

.

Let

f

:

(

X

,

x

0

)

→

(

Y

,

y

0

)

{\displaystyle f\colon (X,x_{0})\to (Y,y_{0})}

be a continuous map between pointed spaces and let

M

f

{\displaystyle Mf}

denote the mapping fibre (the fibration dual to the mapping cone). One then obtains an exact sequence:

M

f

→

X

→

Y

{\displaystyle Mf\to X\to Y}

where the mapping fibre is defined as:

M

f

=

{

(

x

,

ω

)

∈

X

×

Y

I

:

ω

(

0

)

=

y

0

and

ω

(

1

)

=

f

(

x

)

}

{\displaystyle Mf=\{(x,\omega )\in X\times Y^{I}:\omega (0)=y_{0}{\mbox{ and }}\omega (1)=f(x)\}}

Observe that the loop space

Ω

Y

{\displaystyle \Omega Y}

injects into the mapping fibre:

Ω

Y

→

M

f

{\displaystyle \Omega Y\to Mf}

, as it consists of those maps that both start and end at the basepoint

y

0

{\displaystyle y_{0}}

. One may then show that the above sequence extends to the longer sequence

Ω

X

→

Ω

Y

→

M

f

→

X

→

Y

{\displaystyle \Omega X\to \Omega Y\to Mf\to X\to Y}

The construction can then be iterated to obtain the exact Puppe sequence

⋯

→

Ω

2

(

M

f

)

→

Ω

2

X

→

Ω

2

Y

→

Ω

(

M

f

)

→

Ω

X

→

Ω

Y

→

M

f

→

X

→

Y

{\displaystyle \cdots \to \Omega ^{2}(Mf)\to \Omega ^{2}X\to \Omega ^{2}Y\to \Omega (Mf)\to \Omega X\to \Omega Y\to Mf\to X\to Y}

The exact sequence is often more convenient than the coexact sequence in practical applications, as Joseph J. Rotman explains:

(the) various constructions (of the coexact sequence) involve quotient spaces instead of subspaces, and so all maps and homotopies require more scrutiny to ensure that they are well-defined and continuous.

Examples

Example: Relative homotopy

As a special case, one may take X to be a subspace A of Y that contains the basepoint y0, and f to be the inclusion

i

:

A

↪

Y

{\displaystyle i:A\hookrightarrow Y}

of A into Y. One then obtains an exact sequence in the category of pointed spaces:

⋯

→

π

n

+

1

(

A

)

→

π

n

+

1

(

Y

)

→

[

S

0

,

Ω

n

(

M

i

)

]

→

π

n

(

A

)

→

π

n

(

Y

)

→

⋯

⋯

→

π

1

(

A

)

→

π

1

(

Y

)

→

[

S

0

,

M

i

]

→

π

0

(

A

)

→

π

0

(

Y

)

{\displaystyle {\begin{aligned}\cdots &\to \pi _{n+1}(A)\to \pi _{n+1}(Y)\to \left[S^{0},\Omega ^{n}(Mi)\right]\to \pi _{n}(A)\to \pi _{n}(Y)\to \cdots \\\cdots &\to \pi _{1}(A)\to \pi _{1}(Y)\to \left[S^{0},Mi\right]\to \pi _{0}(A)\to \pi _{0}(Y)\end{aligned}}}

where the

π

n

{\displaystyle \pi _{n}}

are the homotopy groups,

S

0

{\displaystyle S^{0}}

is the zero-sphere (i.e. two points) and

[

U

,

W

]

{\displaystyle [U,W]}

denotes the homotopy equivalence of maps from U to W. Note that

π

n

+

1

(

X

)

=

π

1

(

Ω

n

X

)

{\displaystyle \pi _{n+1}(X)=\pi _{1}(\Omega ^{n}X)}

.