Simply-connected 4-manifolds with a given boundary
Authors: Boyer Steven 
Published: July 2010

Let

M
M

be a closed, oriented, connected

3
3

-manifold. For each bilinear, symmetric pairing

(

Z

n

,

L
)

({{\mathbf {Z}}^n},\,L)

, our goal is to calculate the set

V

L

(
M
)

{\mathcal {V}_L}(M)

of all oriented homeomorphism types of compact,

1
1

-connected, oriented

4
4

-manifolds with boundary

M
M

and intersection pairing isomorphic to

(

Z

n

,

L
)

({{\mathbf {Z}}^n},\,L)

. For each pair

(

Z

n

,

L
)

({{\mathbf {Z}}^n},\,L)

which presents

H
∗

(
M
)

{H_ \ast }(M)

, we construct a double coset space

B
L
t

(
M
)

B_L^t(M)

and a function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript upper L Superscript t Baseline colon script upper V Subscript upper L Baseline left-parenthesis upper M right-parenthesis right-arrow upper B Subscript upper L Superscript t Baseline left-parenthesis upper M right-parenthesis">

c
L
t

:

V

L

(
M
)
→

B
L
t

(
M
)

c_L^t:{\mathcal {V}_L}(M) \to B_L^t(M)

. The set

B
L
t

(
M
)

B_L^t(M)

is the quotient of the group of all link-pairing preserving isomorphisms of the torsion subgroup of

H
1

(
M
)

{H_1}(M)

by two naturally occuring subgroups. When

(

Z

n

,

L
)

({{\mathbf {Z}}^n},\,L)

is an even pairing, we construct another double coset space

B
^

L

(
M
)

{\hat B_L}(M)

, a function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove c With caret Subscript upper L Baseline colon script upper V Subscript upper L Baseline left-parenthesis upper M right-parenthesis right-arrow ModifyingAbove upper B With caret Subscript upper L Baseline left-parenthesis upper M right-parenthesis">

c
^

L

:

V

L

(
M
)
→

B
^

L

(
M
)

{\hat c_L}:{\mathcal {V}_L}(M) \to {\hat B_L}(M)

and a projection

p
2

:

B
^

L

(
M
)
→

B
L
t

(
M
)

{p_2}:{\hat B_L}(M) \to B_L^t(M)

such that

p
2

⋅

c
^

L

=

c
L
t

{p_2} \cdot {\hat c_L} = c_L^t

. Our main result states that when

(

Z

n

,

L
)

({{\mathbf {Z}}^n},\,L)

is even the function

c
^

L

{\hat c_L}

is injective, as is the function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c Subscript upper L Superscript t Baseline times normal upper Delta colon script upper V Subscript upper L Baseline left-parenthesis upper M right-parenthesis right-arrow upper B Subscript upper L Superscript t Baseline left-parenthesis upper M right-parenthesis times bold upper Z slash 2">

c
L
t

×
Δ
:

V

L

(
M
)
→

B
L
t

(
M
)
×

Z

/

2

c_L^t \times \Delta :{\mathcal {V}_L}(M) \to B_L^t(M) \times {\mathbf {Z}}/2

when

(

Z

n

,

L
)

({{\mathbf {Z}}^n},\,L)

is odd. Here

Δ
\Delta

is a Kirby-Siebenmann obstruction to smoothing. It follows that the sets

V

L

(
M
)

{\mathcal {V}_L}(M)

are finite and of an order bounded above by a constant depending only on

H
1

(
M
)

{H_1}(M)

. We also show that when

H
1

(
M
;

Q

)
≅
0

{H_1}(M;{\mathbf {Q}}) \cong 0

and

(

Z

n

,

L
)

({{\mathbf {Z}}^n},\,L)

is even,

c
L
t

=

p
2

⋅

c
^

L

c_L^t = {p_2} \cdot {\hat c_L}

is injective. It seems likely that via the functions

c
L
t

×
Δ

c_L^t \times \Delta

and

c
^

L

{\hat c_L}

, the sets

B
L
t

(
M
)
×

Z

/

2

B_L^t(M) \times {\mathbf {Z}}/2

and

B
^

L

(
M
)

{\hat B_L}(M)

calculate

V

L

(
M
)

{\mathcal {V}_L}(M)

when

(

Z

n

,

L
)

({{\mathbf {Z}}^n},\,L)

is respectively odd and even. We verify this in several cases, most notably when

H
1

(
M
)

{H_1}(M)

is free abelian. The results above are based on a theorem which gives necessary and sufficient conditions for the existence of a homeomorphism between two

1
1

-connected

4
4

-manifolds extending a given homeomorphism of their boundaries. The theory developed is then applied to show that there is an

m
>
0

m > 0

, depending only on

H
1

(
M
)

{H_1}(M)

, such that for any self-homeomorphism

f
f

of

M
M

,

f
m

{f^m}

extends to a self-homeomorphism of any

1
1

-connected, compact

4
4

-manifold with boundary

M
M

.