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Addendum to: on Novikov's conjecture for cocompact subgroups of a lie group
Topology and its Applications North-Holland
ADDENDUM COCOMPACT
93
26 (1987) 93-95
TO: ON NOVIKOV’S CONJECTURE SUBGROUPS OF A LIE GROUP
FOR
F.T. FARRELL* Department of Mathematics, Columbia
University,
New York, NY 10027, USA
W.C. HSIANG** Department of Mathematics, Rinceron University, Received
Princeron,
NJ 08544, USA
16 June 1986
Let r be a discrete cocompact torsion-free subgroup of a Lie group G (with finitely many connected components) and K be a maximal compact subgroup of G. If N is a closed manifold homotopically equivalent to the double coset space M = K\G/r, then N xR-’ and M xR3 are homeomorphic. Earlier, the authors had proved this under the extra assumption that G is linear and analytic.
AMS (MOS)
torsion
Subj. Class.:
free discrete
18F25, 22E40, 57R67
subgroups
In this note, we show how to extend the main result of our earlier paper [5] to a slightly more general setting. Using the same notation as in [5], we will prove that Theorem A of [5] remains true when G is an arbitrary Lie group with finitely many components. We had required in [S] that G be a linear analytic group. We start by recalling our notation. Let K be a maximal compact subgroup of G and X = K\G which is diffeomorphic to Euclidean space Iw”. Let rc G be a torsion-free discrete subgroup and M” = X/l- = K\G/l-.
Let 03’ denote the closed i-dimensional 0: [M”xD’,d;
disc and
G/To~,*]+LS,+~(V~.,M”,
w,(M”))
be the surgery map [S, 61 where n + i > 4. The precise statement of the extension claimed above is the following result. Theorem.
Under the above assumptions, 8 is a split monomorphism.
As observed in [2], this Theorem has the following consequence Corollary B of [5]. * Partially ** Partially
supported supported
0166-8641/87/S3.50
by National by National
Science Science
Foundation Foundation
@ 1987, Elsevier
Science
Publishers
Grant Grant
Number Number
which extends
MCS-7923654. GP-39324 Xl.
B.V. (North-Holland)
94
F.T. Farrell, W.C. Hsiang / On Novikov’s
Corollary. equivalence
Let M” =X/f be given as above. where N” is a closed manifold. Then
gxid:
conjecture
Let g: N”+
M”
be a homotopy
N”xR3+MM”xR3
is properly homotopic
to a homeomorphism.
In proving Theorem we may assume, after quotienting G by a compact normal subgroup if necessary, that G does not contain a nontrivial connected compact normal subgroup since this does not change X or M”. (This corrects a minor oversight on page 40 of [5]. Namely, the statement “T, is discrete in L” requires G does not contain a non-trivial connected compact normal subgrdup in order to apply Corollary 8.28 of [7].) Let l+S-+G;L+l
(1)
be the Levi decomposition for G and let KL be a (possibly non-compact) subgroup of L such that L/KL is a symmetric space. Let (as before) K be a maxima1 compact subgroup of G. We may assume that CL(K) c KL. Then the following is a fiber bundle K\$-‘(K,)+
K\G+
KL\L
(2)
where the total space, fiber and base space are diffeomorphic to R” for perhaps different m. We replace the group r, = f fl S of [5] by a group I’s to be constructed presently. First, we list its properties. (9 ri is torsion-free and finitely generated; (ii) Tj contains a virtually nilpotent normal subgroup B such that Ts/B is free abelian; (iii) rg is normal in r; (iv) Ts E $-‘( KL); with compact (v) rg acts freely and properly discontinuously on K\$-‘(K,) quotient. NOWwe construct T.+. Let Z be the center of Lo, where Lo is the connected component of the identity element e in L, then ZE KL. Let K’E KL be the maximal normal compact connected subgroup of L. Let
then Ts is normal in I’. (Tg will be a subgroup with finite index in Ts.) Using a consequence of the Bore1 density theorem, (cf. [7, Corollary 5.171) we can show that Ts acts freely and properly discontinuously on K\$-‘(K,) with compact quotient. Now l’s is normal in TS and C = Tg//Ts is finite by abelian. Consequently, TS is virtually poly-Z. Let N be the nil radical of Ts, then N is a characteristic subgroup of Ts. Let Go be the set of all elements x E C such that (a) x has finite order; and (b) there exists a normal free abelian subgroup A, with finite index in C such that x acts trivially on A, via conjugation.
F.T. Farrell, W.C. Hsiang / On Nouikov’s
Then, Let
G,, is a characteristic
A be the maximal
characteristic
finite subgroup
normal
rs
of C and C = C/G,
free abelian
subgroup
is crystallographic.
of finite index in C, then
A is
in C. Let B be the subgroup
G,c C under the canonical projection A under the canonical projection
Clearly, Redefine
95
conjecture
of Ts which is the inverse image of Ts + C. And let Ts be the inverse image of
satisfies prroperties (i)-(v). S,,, fi” and $h of [5] as follows:
(a) S,= K\V(K,)/rs, (b) An = K\G/l-3, (c) AA= K‘\L. Concatenating the results have that
of [l] and [4] with [3] and using
Y( S, x T’ x D’, a) = 0
provided
dim S,+i
properties
+ i > 4.
Now recall that Theorem A’ of [5] is a general result which is independent Lie group facts. In particular, we have just shown that M” = K\G/T property # of [5]; hence, Theorem A’ applies to M” proving Theorem.
(i)-(v),
we
(3) of special satisfies
References [I] F.T. Farrell and W.C. Hsiang, Manifolds with rr, = Z xp G, Amer. J. Math. 95 (1973) 813-846. [2] F.T. Farrell and W.C. Hsiang, On Novikov’s conjecture for non-positively curved manifolds, I, Ann. of Math. 113 (1981) 199-209. [3] F.T. Farrell and W.C. Hsiang, The Whitehead group of poly-(finite or cyclic) groups, J. London Math. Sot. 24 (1981) 308-324. [4] F.T. Farrell and W.C. Hsiang, Topological characterization of flat and almost flat Riemannian manifolds M” (n #3,4), Amer. J. Math. 105 (1983) 641-672. [S] F.T. Farrell and W.C. Hsiang, On Novikov’s conjecture for cocompact discrete subgroups of a Lie group in: Algebraic Topology Aarhus 1982, Lecture Notes in Math. 1051 (Springer, New York, 1984) 38-48. [6] R.C. Kirby and L.C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Annals of Math. Studies (Princeton University Press, Princeton, 1977). [7] M.S. Raghunathan, Discrete Subgroups of Lie Groups (Springer, New York, 1972). [8] C.T.C. Wall, Surgery on Compact Manifolds (Academic Press, London, 1971).
ADDENDUM COCOMPACT
93
26 (1987) 93-95
TO: ON NOVIKOV’S CONJECTURE SUBGROUPS OF A LIE GROUP
FOR
F.T. FARRELL* Department of Mathematics, Columbia
University,
New York, NY 10027, USA
W.C. HSIANG** Department of Mathematics, Rinceron University, Received
Princeron,
NJ 08544, USA
16 June 1986
Let r be a discrete cocompact torsion-free subgroup of a Lie group G (with finitely many connected components) and K be a maximal compact subgroup of G. If N is a closed manifold homotopically equivalent to the double coset space M = K\G/r, then N xR-’ and M xR3 are homeomorphic. Earlier, the authors had proved this under the extra assumption that G is linear and analytic.
AMS (MOS)
torsion
Subj. Class.:
free discrete
18F25, 22E40, 57R67
subgroups
In this note, we show how to extend the main result of our earlier paper [5] to a slightly more general setting. Using the same notation as in [5], we will prove that Theorem A of [5] remains true when G is an arbitrary Lie group with finitely many components. We had required in [S] that G be a linear analytic group. We start by recalling our notation. Let K be a maximal compact subgroup of G and X = K\G which is diffeomorphic to Euclidean space Iw”. Let rc G be a torsion-free discrete subgroup and M” = X/l- = K\G/l-.
Let 03’ denote the closed i-dimensional 0: [M”xD’,d;
disc and
G/To~,*]+LS,+~(V~.,M”,
w,(M”))
be the surgery map [S, 61 where n + i > 4. The precise statement of the extension claimed above is the following result. Theorem.
Under the above assumptions, 8 is a split monomorphism.
As observed in [2], this Theorem has the following consequence Corollary B of [5]. * Partially ** Partially
supported supported
0166-8641/87/S3.50
by National by National
Science Science
Foundation Foundation
@ 1987, Elsevier
Science
Publishers
Grant Grant
Number Number
which extends
MCS-7923654. GP-39324 Xl.
B.V. (North-Holland)
94
F.T. Farrell, W.C. Hsiang / On Novikov’s
Corollary. equivalence
Let M” =X/f be given as above. where N” is a closed manifold. Then
gxid:
conjecture
Let g: N”+
M”
be a homotopy
N”xR3+MM”xR3
is properly homotopic
to a homeomorphism.
In proving Theorem we may assume, after quotienting G by a compact normal subgroup if necessary, that G does not contain a nontrivial connected compact normal subgroup since this does not change X or M”. (This corrects a minor oversight on page 40 of [5]. Namely, the statement “T, is discrete in L” requires G does not contain a non-trivial connected compact normal subgrdup in order to apply Corollary 8.28 of [7].) Let l+S-+G;L+l
(1)
be the Levi decomposition for G and let KL be a (possibly non-compact) subgroup of L such that L/KL is a symmetric space. Let (as before) K be a maxima1 compact subgroup of G. We may assume that CL(K) c KL. Then the following is a fiber bundle K\$-‘(K,)+
K\G+
KL\L
(2)
where the total space, fiber and base space are diffeomorphic to R” for perhaps different m. We replace the group r, = f fl S of [5] by a group I’s to be constructed presently. First, we list its properties. (9 ri is torsion-free and finitely generated; (ii) Tj contains a virtually nilpotent normal subgroup B such that Ts/B is free abelian; (iii) rg is normal in r; (iv) Ts E $-‘( KL); with compact (v) rg acts freely and properly discontinuously on K\$-‘(K,) quotient. NOWwe construct T.+. Let Z be the center of Lo, where Lo is the connected component of the identity element e in L, then ZE KL. Let K’E KL be the maximal normal compact connected subgroup of L. Let
then Ts is normal in I’. (Tg will be a subgroup with finite index in Ts.) Using a consequence of the Bore1 density theorem, (cf. [7, Corollary 5.171) we can show that Ts acts freely and properly discontinuously on K\$-‘(K,) with compact quotient. Now l’s is normal in TS and C = Tg//Ts is finite by abelian. Consequently, TS is virtually poly-Z. Let N be the nil radical of Ts, then N is a characteristic subgroup of Ts. Let Go be the set of all elements x E C such that (a) x has finite order; and (b) there exists a normal free abelian subgroup A, with finite index in C such that x acts trivially on A, via conjugation.
F.T. Farrell, W.C. Hsiang / On Nouikov’s
Then, Let
G,, is a characteristic
A be the maximal
characteristic
finite subgroup
normal
rs
of C and C = C/G,
free abelian
subgroup
is crystallographic.
of finite index in C, then
A is
in C. Let B be the subgroup
G,c C under the canonical projection A under the canonical projection
Clearly, Redefine
95
conjecture
of Ts which is the inverse image of Ts + C. And let Ts be the inverse image of
satisfies prroperties (i)-(v). S,,, fi” and $h of [5] as follows:
(a) S,= K\V(K,)/rs, (b) An = K\G/l-3, (c) AA= K‘\L. Concatenating the results have that
of [l] and [4] with [3] and using
Y( S, x T’ x D’, a) = 0
provided
dim S,+i
properties
+ i > 4.
Now recall that Theorem A’ of [5] is a general result which is independent Lie group facts. In particular, we have just shown that M” = K\G/T property # of [5]; hence, Theorem A’ applies to M” proving Theorem.
(i)-(v),
we
(3) of special satisfies
References [I] F.T. Farrell and W.C. Hsiang, Manifolds with rr, = Z xp G, Amer. J. Math. 95 (1973) 813-846. [2] F.T. Farrell and W.C. Hsiang, On Novikov’s conjecture for non-positively curved manifolds, I, Ann. of Math. 113 (1981) 199-209. [3] F.T. Farrell and W.C. Hsiang, The Whitehead group of poly-(finite or cyclic) groups, J. London Math. Sot. 24 (1981) 308-324. [4] F.T. Farrell and W.C. Hsiang, Topological characterization of flat and almost flat Riemannian manifolds M” (n #3,4), Amer. J. Math. 105 (1983) 641-672. [S] F.T. Farrell and W.C. Hsiang, On Novikov’s conjecture for cocompact discrete subgroups of a Lie group in: Algebraic Topology Aarhus 1982, Lecture Notes in Math. 1051 (Springer, New York, 1984) 38-48. [6] R.C. Kirby and L.C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Annals of Math. Studies (Princeton University Press, Princeton, 1977). [7] M.S. Raghunathan, Discrete Subgroups of Lie Groups (Springer, New York, 1972). [8] C.T.C. Wall, Surgery on Compact Manifolds (Academic Press, London, 1971).