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automorphism h" " GL(m, Z) is trivial. Hence in the lifted sequence, the semi-
direct product Zm Zp is a direct product. But, this contradicts admissibility of
m s m s m
T . So Ty = 1 for each y " T . That is, the action of T is free on T .
s
Moreover, we claim it is a product action. Now put H = Im(evx(À1(T )))
"
m s m
and Q = À1(T )/H Zm/Zs. Q acts properly on T \TH = W , a contractible
=
(cohomology) manifold. Then, Q, which is abelian, must act freely since Qw = 1 for
s s
every w " W by Qw
=
s
this means Q is torsion free, by Corollary 1.15.4 . So Im(evx(T )) Zs is a direct
=
"
summand of Zm. This will force the action to split as we have seen earlier in
s m s s
Theorem 1.14.2, so (T , T ) = (T , T × Y ) where Y is a (cohomology) manifold
m-s
of the homotopy type of T .
Now F acts on Y , and acts trivially on À1(Y ), a summand of Zm. In the lifting
sequence 1 ’! À1(Y ) ’! E ’! F ’! 1, the group E acts trivially on À1(Y ) because G
m
acts trivially on À1(T ). Thus, E is a central extension of À1(Y ) Zm-s and since
=
it maps trivially into GL(m - s, Z), it must be torsion free. Otherwise, E would
contain an effective action of À1(Y ) × Zp, for some prime p, on Y , a contractible
(cohomology) manifold with compact quotient which is impossible. The Theorem
?? was proven for topological manifolds, but it is also true by the same argument for
ANR cohomology manifolds. Then Y is an ANR aspherical cohomology manifold
homotopy equivalent to (m - s)-torus.
[If the action is smooth (respectively, locally smooth), then Y is a smooth
m-s
manifold (respectively, topological manifold), with the homotopy type of T .
m-s
In particular, using standard surgery results, Y will be homeomorphic to T
provided m - s = 3. For some pathological topological actions, Y could fail to be
locally Euclidean.]
c&
Therefore, E as we shall see later in () is isomorphic to Zm-s, and F is abelian, ???
Moreover, as it must act properly and freely, E acts as covering transformations on
s
Y . Actually we can now see the structure of G as T × F where F is an abelian
m-s
group isomorphic to a subgroup of T . As G splits as a product, we let F act on
m
T . Then it acts as covering transformations since it does so on the factor Y . Of
m s
course, F \T is again a topological torus. The T action can be treated as acting
m m
on T or F \T .
This result for the torus has a generalization to infra-nilmanifolds and will be
treated in Chapter 10. Is this right??
old b-maxtorus.tex taked out [0.07]
1.17.16 Example. There are no shortage of closed aspherical manifolds. For exam-
ple, if “ is discrete and acts properly and freely on Rn, then “\Rn is an aspherical
manifold (a K(“, 1)-manifold). If “ acts so that the quotient is compact, then “\Rn
is a closed aspherical manifold. Typical examples arise by taking “ a torsion free
cocompact discrete subgroup of a Lie group G with finitely many components and
K a maximal compact subgroup of G. Since K is a maximal compact subgroup,
G/K is diffeomorphic to Rn for some n, and “ being torsion free implies “ )" K = 1
so that “\Rn = “\(G/K) = (“\G)/K is a closed aspherical manifold.
50 1. TRANSFORMATION GROUPS
For example, each closed 2-manifold whose Euler characteristic is negative is of
the form “\PSL(2, R) Z2/O(2) where “ is the fundamental group of the surface and
is isomorphic to a torsion free cocompact subgroup of the full group of isometries
of the hyperbolic plane.
Hyper-aspherical manifolds (see section ??) which are not aspherical are most
easily obtained by taking any closed oriented aspherical n-manifolds N and forming
the orientable connected sum with any other closed oriented m-manifold P . Then
M = P #N maps onto N with a map of degree 1 by collapsing P - (Ball)0 ‚" M to
a point.
Each of the implications of Theorem ?? cannot be reversed. See [?] for a
complete discussion. For example, the connected sum of two non-homeomorphic
3-dimensional spherical space forms which are also not lens spaces is admissible but
is not a K-manifold.
1.17.17. There are many interesting examples of closed aspherical manifolds without
any compact Lie group action. The first examples were given by Conner Raymond
Weinberger, and E. Bloomberg in his thesis. The first 3-dimensional examples were
Put in Refs given by Raymond Tollefson.
In every bordism class (dim e" 3), there exist such manifolds. This was proved [ Pobierz całość w formacie PDF ]