Let
T
O
P
(
M
)
TOP(M)
be the simplicial group of homeomorphisms of
M
M
. The following theorems are proved. Theorem A. Let
M
M
be a topological manifold of dim
≥
\geq
5 with a finite number of tame ends
ε
i
\varepsilon _{i}
,
1
≤
i
≤
k
1\leq i\leq k
. Let
T
O
P
e
p
(
M
)
TOP^{ep}(M)
be the simplicial group of end preserving homeomorphisms of
M
M
. Let
W
i
W_{i}
be a periodic neighborhood of each end in
M
M
, and let
p
i
:
W
i
→
R
p_{i}: W_{i} \to \mathbb {R}
be manifold approximate fibrations. Then there exists a map
f
:
T
O
P
e
p
(
M
)
→
∏
i
T
O
P
e
p
(
W
i
)
f: TOP^{ep}(M) \to \prod _{i} TOP^{ep}(W_{i})
such that the homotopy fiber of
f
f
is equivalent to
T
O
P
c
s
(
M
)
TOP_{cs}(M)
, the simplicial group of homeomorphisms of
M
M
which have compact support. Theorem B. Let
M
M
be a compact topological manifold of dim
≥
\geq
5, with connected boundary
∂
M
\partial M
, and denote the interior of
M
M
by
I
n
t
M
Int M
. Let
f
:
T
O
P
(
M
)
→
T
O
P
(
I
n
t
M
)
f: TOP(M)\to TOP(Int M)
be the restriction map and let
G
\mathcal {G}
be the homotopy fiber of
f
f
over
i
d
I
n
t
M
id_{Int M}
. Then
π
i
G
\pi _{i} \mathcal {G}
is isomorphic to
π
i
C
(
∂
M
)
\pi _{i} \mathcal {C} (\partial M)
for
i
>
0
i > 0
, where
C
(
∂
M
)
\mathcal {C} (\partial M)
is the concordance space of
∂
M
\partial M
. Theorem C. Let
q
0
:
W
→
R
q_{0}: W \to \mathbb {R}
be a manifold approximate fibration with dim
W
≥
W \geq
5. Then there exist maps
α
:
π
i
T
O
P
e
p
(
W
)
→
π
i
T
O
P
(
W
^
)
\alpha : \pi _{i} TOP^{ep}(W) \to \pi _{i} TOP(\hat W)
and
β
:
π
i
T
O
P
(
W
^
)
→
π
i
T
O
P
e
p
(
W
)
\beta : \pi _{i} TOP(\hat W) \to \pi _{i} TOP^{ep}(W)
for
i
>
1
i >1
, such that
β
∘
α
≃
i
d
\beta \circ \alpha \simeq id
, where
W
^
\hat W
is a compact and connected manifold and
W
W
is the infinite cyclic cover of
W
^
\hat W
.