OBSTRUCTIONS TO FIBERING A MANIFOLD 7
we ca n equip E
n
with the pushout simple structure. Lemma 2.3 (ii) implies that
the choice of T
i
does not matter.
Notice that the choice of the characteristic maps (Q
i
, q
i
) does not belong to the
structure of a CW-complex. Only the skeletal filtration (X
n
)
n≥−1
is part of the
structure and the existence of a pushout as above is required but not specified. One
can recover the open cells by the path-components of X
n
−X
n−1
and the closed cells
by the closure of the open cells but not the characteristic maps Q
i
. Therefore one
has to show that the simple structure on E
n
is independent of the choice of these
pushouts. This is done by thickening X
n−1
into X
n
. The details of the argument
are similar to the o ne given in the proof of [15, Lemma 7.13] and can be found
in [23, Subsection 3.2].
Remark 3.3. If p is triv ial, i.e., p : B × F → B is the projection map, and F is a
finite CW -complex, then for any spider s, the simple structure ξ(b, s, ξ
can
(F )) on
B × F agrees with the product simple structure.
The dependence o f the simple structure on the choice of (b, s, ζ) is described in
the next lemma. Therefore suppose that another cho ice (b
′
, s
′
, ζ
′
) has been made,
with b
′
∈ B, s
′
a spider at b
′
, and ζ
′
a simple structure on the fiber F
b
′
.
Lemma 3.4. S uppose that B is path-connected. Given a cell c ∈ I(B), let u
c
be
any path in the interior of c from w
c
(1) to w
′
c
(1), where w
c
and w
′
c
are given by the
spiders s and s
′
, and let v
c
be the path w
c
∗ u
c
∗ (w
′
c
)
−
. Then the homotopy class
relative endpoints [v
c
] is independent of u
c
. If we denote by (i
b
′
)
∗
: Wh(π(F
b
′
)) →
Wh(π(E)) the homomorphism induced by the inclusion i
b
′
: F
b
′
→ E, the following
holds in Wh(π(E))
τ
(E, ξ(b, s, ζ))
id
−→ (E, ξ(b
′
, s
′
, ζ
′
))
=
X
c∈I(B)
(−1)
dim(c)
· (i
b
′
)
∗
τ
(F
b
, ζ)
t([v
c
])
−−−−→ (F
b
′
, ζ
′
)
Proof. This follows from Lemma 1.4 and Lemma 2.3.
Let p : E → B be a fibration whose fiber has the homotopy type of a finite
CW -complex. We can assign to it a class
Θ(p) ∈ H
1
B, Wh(π(E))
(3.5)
as follows. For simplicity we assume that B is path-connected. Given b ∈ B, a loop
w at b in B and a simple structure ζ on F
b
, we can compute the Whitehead torsion
of the fiber trans port along w
(i
b
)
∗
τ
t([w]): (F
b
, ζ) → (F
b
, ζ)
∈ Wh(π(E))
for i
b
: F
b
→ E the inclusion. From Lemma 1.4 and Lemma 2.1 one co ncludes that
this element in independent of the choice of ζ and that we obtain a group homo-
morphism π
1
(B, b) → Wh(π(E)). It defines an element Θ(p) ∈ H
1
B; Wh(π(E))
which is independent of the choice of b ∈ B.
Definition 3.6. Let p: E → B be a fibration whose fiber has the homotopy type
of a finite CW -complex. We call p simple if Θ(p|
C
) = 0 holds for any component
C ∈ π
0
(B) with respect to the restriction E|
C
→ C.
Lemma 3.7. Let p: E → B be a locally trivial fiber bundle with a finite CW -
complex as typical fiber and paracompact base space. Then it is a simple fibration.
Proof. It is a fibration by [27, page 33]. It is simple, since the fiber transport
in such a bundle is given by homeomorphisms and the Whitehead torsion of a
homeomorphism is trivial (see [2]).