The topological structure of 4-manifolds with effective torus actions. I
Authors: Pao Peter Sie 
Published: July 2010

Torus actions on orientable 4-manifolds have been studied by F. Raymond and P. Orlik [8] and [9]. The equivariant classification problem has been completely answered there. The problem then arose as to what can be said about the topological classification of these manifolds. Specifically, when are two manifolds homeomorphic if they are not equivariantly homeomorphic? In some cases this problem was answered in the above mentioned papers. For example, if the only nontrivial stability groups are finite cyclic, then the manifolds are essentially classified by their fundamental groups. In the presence of fixed points, a connected sum decomposition in terms of

S
4

,

S
2

×

S
2

,
C

P
2

,
−
C

P

−
2

,

S
1

×

S
3

{S^4},{S^2} \times {S^2},C{P^2}, - C{P^{ - 2}},{S^1} \times {S^3}

, and three families of elementary 4-manifolds,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R left-parenthesis m comma n right-parenthesis comma upper T left-parenthesis m comma n semicolon m prime comma n Superscript prime Baseline right-parenthesis comma upper L left-parenthesis n semicolon p comma q semicolon m right-parenthesis">

R
(
m
,
n
)
,
T
(
m
,
n
;

m
′

,

n
′

)
,
L
(
n
;
p
,
q
;
m
)

R(m,n),T(m,n;m’,n’),L(n;p,q;m)

has been obtained (where m, n,

m
′

,

n
′

m’,n’

, p, and q are integers). In addition, a stable homeomorphic relation for the manifolds

R
(
m
,
n
)

R(m,n)

and

T
(
m
,
n
;

m
′

,

n
′

)

T(m,n;m’,n’)

can also be found in [9]. But the topological classification of R’s, T’s, and especially L’s were still unsolved problems. Furthermore, the connected sum decomposition of a manifold with fixed points, even in the simply connected case, is not unique. In this paper, we completely classify the manifolds with fixed points. For the manifolds R’s and T’s, the above mentioned stable homeomorphic relation is proved to be the topological classification. The manifolds

L
(
n
;
p
,
q
;
m
)

L(n;p,q;m)

form a very interesting family of 4-manifolds. They behave similarly to lens spaces. For example, the fundamental group of

L
(
n
;
p
,
q
;
m
)

L(n;p,q;m)

is finite cyclic of order n. And it is proved that

π
1

(
L
(
n
;
p
,
q
;
m
)
)

{\pi _1}(L(n;p,q;m))

and

π
1

(
L
(
n
;

p
′

,

q
′

;

m
′

)
)

{\pi _1}(L(n;p’,q’;m’))

act identically on the second cohomology of their common universal covering space,

(

S
2

×

S
2

)
#
⋯
#

(

S
2

×

S
2

)

({S^2} \times {S^2})\# \cdots \# \;({S^2} \times {S^2})

(

n
−
1

n - 1

copies), even though

L
(
n
;
p
,
q
;
m
)

L(n;p,q;m)

and

L
(
n
;

p
′

,

q
′

;

m
′

)

L(n;p’,q’;m’)

are not homotopically equivalent for some

(
n
;

p
′

,

q
′

;

m
′

)

(n;p’,q’;m’)

’s. This family of manifolds is explicitly constructed and completely classified. In addition, a normal form is imposed on a connected sum decomposition mentioned above. These nromal forms are unique. Most of the material of this paper appeared first in the author’s doctoral dissertation. The author would like to thank his thesis advisor Professor F. Raymond for his help and encouragement.