Orientation reversal of manifolds - Universität Bonn

34 3 Examples in every dimension ≥ 3
Lemma 28
Let f ∶ T n → T n be an orientation-preserving diffeomorphism such that the
difference f ∗ − id is surjective on H 1 (T n ). Let T ∶ M f → M f be a continuous
map. The induced map T ∗ on H n+1 (M f ) ≅ Z is given by
det((T ∗ ) ∣N ) ⋅ det((T ∗ ) H ),
where in our case, N = π 1 (T n ) ≅ Z n and H = π 1 (S 1 ) ≅ Z.
Proof. The strategy is to show that T is homotopic to a fibre-preserving map
and then to exploit naturality of the Serre spectral sequence.
Consider the diagram
M f
p
T
M f
p
S 1 t S 1
where p is the projection in our fibre bundle and t has to be defined.
Since there is a natural bijection [M f , S 1 ] ≅ H 1 (M f ), which is isomorphic
to Z, this diagram commutes up to homotopy, with the map t being any
(basepoint-preserving) map with the correct degree.
By the homotopy lifting property, T is homotopic to a fibre-preserving map
(and still preserving the basepoint), so we can replace T w. l. o. g. by this map.
Now we are in the situation of a commutative diagram
T n p
M f
T ∣T n T
T n M f
p
S 1 t
S 1
so that we can apply the naturality of the Serre spectral sequence. To be precise,
we consider the E 2 term of the homology spectral sequence for the fibration p.
The only term with total degree at least n + 1 is E1,n 2 = H 1(S 1 ; H n (T n )). A priori,
the coefficients are local, but since we specified f ∶ T n → T n as orientationpreserving,
the coefficient group is in fact constant. Since there are no differentials
from or to E1,n 2 , we have E2 1,n = E∞ 1,n . Since there are no other terms in
degree n + 1, we have a natural isomorphism H 1 (S 1 ; H n (T n )) ≅ H n+1 (M f ) ≅ Z.
The word “natural” here refers to fibre-preserving maps of M f , as always in the
context of the Serre spectral sequence. Note that the map t ∗ ∶ H 1 (S 1 ) → H 1 (S 1 )
coincides with (T ∗ ) H . Furthermore, (T ∣T n) ∗ = (T ∗ ) ∣N is given by the determinant
of the map on π 1 (T n ) ≅ Z n as is proved by the cohomology product
structure of the n-torus.
The induced map on H 1 (S 1 ; H n (T n )) is the tensor product of the two maps
above, hence the lemma is proved.