Algebraic Topology and Transformation Groups: Proceedings of a Conference held in Göttingen, FRG, August 23–29, 1987

298 Pages·1988·10.711 MB·English

by  Stefan Bauer (auth.), Tammo tom Dieck (eds.)

The Homotopy Type of a 4-Manifold
with finite ~"Sandamental Group
by Stefan Bauer*
ABSTRACT: ... is determined by its quadratic 2-type, if the 2-Sylow subgroup has
4-periodic cohomology.
The homotopy type siomfp ly connected 4-manifolds si determined by the intersection
form. This is a well-known result of J.H.C. Whitehead and .3 Mitnor. In the non-simply
connected case the homotopy groups lr~ and ~7 and the first k-invariant k E H3(71, v2)
give other homotopy invariants. The quadratic 2-type of an oriented closed 4-manifold
is the isometry class of the quadruple [71(M), ~r2(M), k(M),~/(~)], where 7(It:/) denotes
the intersection form on 72(M) = H2(M). ~ An isometry of two such quadruples is an
isomorphism of 17 and 27 which induces an isometry on 7 and respects the k-invariant.
Recently [H- K] I. Hambleton and .M Kreck, studying the homeomorphism types of
4-manifolds, showed that for groups with periodic cohomology of period 4 the quadratic
2-type determines the homotopy type.
This result can be improved away from the prime .2
Theorem: Suppose the 2-Sylow subgroup of G has 4-periodic cohomology. Then the
homotopy type of an oriented 4-dimensional Poincarfi complex with fundamental group G
is determined by its quadratic 2-type.
I am indebted to Richard Swan for showing me proposition .6 Furthermore I am
grateful to the department of mathematics at the University of Chicago for its hospitality
during the last year.
* Supported by the DFG
Let X be an oriented 4-dimensional Poincar6 complex with finite fundamental group,
f : X ~-- B its 2-stage Postnikov approximation, determined by ~rl, ,2r7 and k, a.nd let ~,(X)
denote the intersection form on /72(2). Then S PD (B, 7(X)) denotes the set of homotopy
types of 4-dimensional Poincar~ complexes Y, together with 3-equivalences g : Y *--- B,
such that f and g induce an isometry of the quadratic 2-types.
The universal cover /) is an Eilenberg-MacLane space and hence, by [MacL], H4(/))
P(~r2(B)), the ZTh(B)-modute F(Tr2(B)) being the module of symmetric 2-tensors, i.e. the
kernel of the map (1 - T): r~ 2 (B) ® 2r7 (B) -* ~c7 (B) ® 2r7 (B), (1 - 7-)(a ® b) = a - ® b ® b a.
The intersection form on 2 sdnopserroc to L[2] of the fundamental class 12[ • Hal2; z).
Let/:/. denote Tate homology.
Proposition 1: If X is a Poincarfi space with finite fundamental group G, then there
is .a bijection [Io( G; (ar~ X) ) , , sPD ( B, (y" X)).
The proof uses a lemma of [H-K]:
Lemma 2: Let, (X,f) and (Y,g)be elements in sPD(B, 7(X)). Then the only
obstruction for the existence of a homotopy equivalence h : X ~ Y over B is the vanishing
ofg.[Y] - f.[X] • H4(B).
Lemma 3: Given a diagram
Z ~ M
Z
such that the torsion in the cokernel of ~ is annihilated by n, then the torsion subgroup
in the pushout K is isomorphic to the torsion subgroup of coker(c~).
Proof of 3: Since the torsion subgroup of M maps injectively into K as well as into
coker(a), we may assume it trivial. Then M is isomorphic to NO < z > with a(1) = mz
for an integer m dividing n. The pushout then is isomorphic to (N@Z@Z)/< (0, m, n) >~
M • Z/m. &
Proof of proposition 1: Let (X, f) and (Y, g) be elements in SPD(B) such that f
and g induce an isometry of the quadratic 2-types. Let 7(X) = "y(Y) = dye"note the inter-
section form on H2(X )~ and H2(]'z). By ]W[ one has ~r3(X) - F(~r2(X))/(7) -- H4(/~,)~ )~
and ~ra(X) ®zv Z ~ Ha(B, X). In the pushout diagramm:
0 0 0
o ~ Ha(2) ®za Z ~ Ha(/)) ®~ Z , Ha(/), 2) ®zc Z , 0
¢ 1 1 1 -~
0 ~ H4(X) , Ha(B) .... ~ Ha(B, X) 0
.J ~. +
0 ~ Ha(X, 2) ;-~ Ha(B,~)) , 0
0 0
the torsion subgroup of H4(B, X) is isomorphic to the torsion subgroup of H4(B) by lemma
3: The module Ha(/), 2) is torsion free. Hence the torsion subgroup of Ha(/), 2) ®zc Z
is annihilated by the order n of the group G. Note that ¢ is just multiplication by n. In
particular one has
Torsion(Ha(B)) -~ Torsion(Ha(B, X)) ~ [/o(G;Tra(X))
Since X and Y have the same quadratic 2-type, (cid:127)[X] = ~.[1~], hence we have
L[X] - g.[~'] • To~,io,~(HaB). This gives an injection
sPD( B, 7) ~-~/:/o(G;Trs(X)).
What about surjectivity? Let K C "A denote a subspace, where one single orbit is deleted.
Let a e ~ra(K) map via the surjection ~r3(K) *-- 7ra(X) *-- ~ra(X) ®zc Z to a given element
& 6 f/o(G;~rs(X)). Let ¢f be the image of 1 6 ZG ~ tIa(f(, K) -~ 7r4(2, K) *--' 7ra(K).
Now let k : 3 S *--- K represent c~+fl and define X~ := (KUk(Gx Da))/G. One has
to show that X~ is an orientable Poincar4 space. Orientability is clear, since Ha(X~) -~
Ha(X~,K) -~ Z. Let f : Xa *-- B extend fIK/a. The intersection form on )~ is
determined by
.)2(aH
.~.[Xc,] - trf(f,~.[X~]) e
But we have fa.[Xa] = f,[X] + :~o In the following diagram 1 6 Z ~ 7ra(X, B) is mapped
to f,[X] e Ha(B).
H4(X) ~ H4(X,K/G) ~,-- Ha(X,K) ~' m(X,K) --, ~ra(K)
ft l 1 l l=
H~(B) ~ = H~(B, K/a) ,--~ H~([3, K) ~ = ~(B, K) -~ ,~(K)
If the upper row is replaced by the corresponding row for X~ and the vertical maps by
the ones induced by f~, then t E ZG is mapped (counterclockwise) to f~.[X~] on the one
hand, on the other hand (clockwise) to f.[X] + .~o
Since the torsion element ~o lies in the kernel of the transfer, one immediately gets
/o.[x~] = f.[x].
In the sequel all ZG-modules have underlying a free abelian group.
The short exact sequence
0 ~ Z ~-L r(~X) , ~3(X) *-- 0
gives rise to an exact sequence in Tate homology:
H0(a;Z) , H0(C;r(~X)) , H0(a;~3(X)) ~ H_I(C;Z) ~, H_~(a;r(~x))
Here/:/0(G; Z) = 0 and /:/_I(G;Z) ~ Z/I Gl.The sequence above gives the connection to
[H-K], theorem(].l).
In order to analyze this sequence, I recall some facts from [H-K],§g2 and 3.
Facts:
)1 F(ZG)= (~i Z[G/Hi] ® F, ,,,here the summation is over all subgroups Hi of
order 2 and F is a free ZG-module.
2) r(zG) r([) • za r(z*) • ZG.Here I denotes the augmentation ideal,
I* its dual.
3)
The modules D3Z and SaZ are (stably!) defined by exact sequences
O--~QsZ-~ F2-~ Z--+O Fo--~ FI..-~
dna
+--O Z ~ IF ~ 2F ~ 3F ~ S3Z ~ O
modules with free .,F
There si na exact ecneuqes
0 ~ f~az , ~r~(X) • rZG ~ S3Z ~ 0
Lemma 4: If 0 *--- A ~-- B *-- C ~ 0 is a short exact sequence of ZG-modules, which
are free over Z, then there are short exact sequences
0~P(A) , F(B) , ,,,D~0
and
0~A®zC ,D---*F(C) ~0.
Proofi Given Z-bases {ai}, {c/} and {ai,~j} of A, C and B, the map h : a, @c/ *---
ai ® ci + Jc ® ai is well-defined and equivariant modulo F(A). &
To prove the theorem, it suffices to show that H0(G; 7r3(X)) = 0. This in turn can be
done separately for each p-Sylow subgroup Gp of G.
Proposition 5: The map 7. : H-l(Gp;Z)~ /-/-l(@; F(,r2(X))) is injective, if
either p is odd or resg 7r2(X) ~ A @ B splits such that the rank of B over Z is odd. In
general the kernel is at most of order 2.
Proofi For the sake of brevity, let r7 denote 7r2(X) and also let F denote the module
r(Tr). Now look at the following sequence of maps:
¢: Z ,Y" P ~ r~ @ r~ ~ gom(Tr', ~r), a *~ gom(~r, )r~ trace Z.
A genera.tor of Z is mapped in Hom(Tr*, ~) to the Poincar6 map a : *r7 -- H2(2) --~
H:(2) = rr, and then to the element id e 7:). Horn(or, So we have ¢(1) = rankz(Tr).
Fact 3) gives rankz(Tr) -= -2 mod G h I hence the i~duced selfmap ¢, of Z/I@I
_/-fa ~(@; Z) is multiplication by -2. This proves, that the kernel is at most of order 2. In
particular it is trivial, if p is odd.
In case p = 2 and resaa rr ~- A @ B, such that the rank of the underlying group
of B is odd, one can replace the map IIom(rr, )r7 t2-~ ~ Z by the map Hom(rr, )rT ,..i*
Horn(B, B) tra,¢ Z in the defining sequence for ¢. A similar argument as above for p odd
gives the claim. &
Remark: The module resa G ~r~(X) ahvays splits, if H4(G;Z) ~ Ext~a(SaZ,fl3Z)
has no 2-torsion, in particular if ~2 has 4-periodic cohomology.
Proposition 6: Let A denote either ~'~Z or S'~Z and let r be the selfmap of A ® A
which permutes the factors. Then (-1)n~ - induces the identity on [/0(G;A N A).
Proofi Let F. ~-- Z be a free resolution of Z and let .F_ be the truncated complex with
Fi = Fi for i < n - 1, ~',, = f/'~ and/~ = 0 else. There is an obvious projection f: F. --~/~'.,
such that fn = 0n. The tensor product F.®F. ~ = F. again is a free resolution of Z and/~.2
is a truncatedfree resolution of Z with/v~,~ = f~Z @ OZ. The chain map f N f induces an
isomorphism of H. 2.'-,/( @z a Z) and H.(F. - 2 @zaZ) in the dimensions * <_ 2n. The selfmap
of F~., as usual defined by t(z ® y) (--1)deg(x)deg(y)x ® y, is a chain automorphism,
=
inducing the identity on the augmentation, hence on all derived functors, in particular on
.2.~/
H. 2 (/7. ®zc Z) = H. (G;Z ). In the same way an involution t can be defined on and
f ® f commutes with t. Obviously ~;,, = (--1)'~r. Hence (-1)'%" induces the identity on
::o(a;z).
®zc z) =
The proof for S ~' Z is dual. &
Proof of the theorem: By proposition ,1 it suffices to show that /t0(G;za(X))
vanishes. By proposition 4 and the remark following it, this group is isomorphic to
/;/0(G; ~rT(F (X))). In order to show that this group vanishes it suffices, by lemma 3, to show
that [to(G;A) vanishes for A e {['(aaZ),F(SaZ),f23Z ® SaZ) But /I0(G;aaZ ® SaZ) -
f/0(G;Z) = 0. Given a module B (with underlying free abelian group), there is a short
exact sequence
0 .--, F(B) --, B ® B .., A2(B) +--- 0.
The map % which flips the both factors, induces, if applied to B E {f~aZ,S3Z} the
following diagram:
*-- /)I(G;A(B)) ~ !flo(G;r(B)) ~ Ho(G;B®B) -*
I (-id) I ie ~ (-id)
The right vertical map is (-id) by proposition 5. This diagram shows that any element
H0(G;I'(B))
in is annihilated by 4.In particular this group vanishes, if G is a p-group for
an odd prime p. That //0(G2; F(B)) vanishes, if ~G has 4-periodic cohomology, follows at
once from the facts 1 - 3, since in this case f'taZ = I* @ nZG and SaZ = I @ nZG
Z/2
Final Remark: An elementary but lengthy computation shows F(SaZ) - Z/2e
fIo(Z/2®Z/2;F(~3Z®S3Z))
and F(f~3Z) = 0 for G = Z/2eZ/2. In particular the group
si nontrivial. Hence the argument above won't work in general.
REFERENCES
B[ ]1
K.S. Brown: CohomoIogy of groups. GTM 87, Springer-Verlag, N.Y. 1982
R. Brown: Elements of Modern Topology. McGraw- Hill, London, 1968
]K-H[
I. Hambleton and M. Kreck: On the Classification of Topological 4-Manifolds
with finite Fundamental Group. Preprint, 1986
[MacL] S. MacLane: Cohomology theory of abelian groups. Proc. Int. Math. Congress,
vol. 2 (1950), pp 8 - 41
]w[
J.H.C. Whitehead: On simply connected 4-dimensionM polyhedra. Comment.
Math. Helv., 22 (1949), pp 81. - 92.
Sonderforschungsbereich 170
Geometrie und Analysis
Mathematisches Institut
Bunsenstr. 3 - 5
D-3400 GSttingen, FRG
Rational Cohomology of
Configuration Spaces of Surfaces
C.-F. B6digheimer and F.R. Cohen
I. Introduction. The k-th configuration space ck(M) of a manifold M is
the space of all unordered k-tuples of distinct points in M. In previous
work [BCT] we have determined the rank of H.(ck(M) ;~ ) for various
fields ~ . However, for even dimensional M the method worked for F =F 2
only. The following is a report on cal~ulations of H*(ck(M) ;Q) for M
a deleted, orientable surface. This case is of considerable interest
because of its applications to mapping class groups, see [BCP].
Similar results for (m-1)-connected, deleted 2m-manifolds will appear
in [BCM].
2. Statement of results. The symmetric group k I acts freely on the space
~k(M) of all ordered k-tuples z( I ., ..,z , k) zi6M, such that z± # z; for
# j. i The orbit space is ck(M). As in [ BCT] we will determine the
rational vector space H~(ck(M) ;Q) as part of the cohomology of a much
larger space. Namely, if X is any space with basepoint x o, we consider
the space
where I, (z .... Zki;Xl, .... k) ~ x z( I ..... zn_1;x I ..... Xk_ )I if k x :x o-
The space C is filtered by subspaces
lkI[
Cj j x
)2( FkC(M;X) = [j--_j\ )M( ~j ) /~
and the quotients FkC/Fk_IC are denoted by Dk(M;X) -
Let M denote a closed, orientable surface of genus g, and M is
g g g
minus a point. We study C(Mg;S 2n) for nZl. ~ H will always stand for
8
rational cohomology, and [P ] resp. E[ ] for polynomial resp.
exterior algebras over ~.
Theorem A. There is an isomorphism of vector spaces
)3( HwC(Mg;S 2n) ~ P[v,u I ..... U2g]®H~(E[w,z ,I .... Z2g ],d)
with Ivi=2n, iuiJ=4n+2, lwl=4n+1, Izil=2n+1, and the differential
d ss_i @iven y_b d(w) = 2(ZlZ + z ... + 2 2g-lZ2g ")
Giving the generators weights, wght )v( = wght(zi) I = and wght(u )i
=wght(w) = 2, makes H C into a filtered vector space. eW denote this
weight filtration by FkH~C. The length filtration FkC of C defines a
second filtration H FkC of H C.
Theorem B. As vector spaces
2n e s2n)
)4( H FkC(Mg;S ) =FkH C(Mg; .
It follows that H Dk(Mg;S 2n) is isomorphic to the vector subspace
of H~(g,n) = P[v,u i] ®H (E[w,zi],d) spanned by all monomials of weight
exactly k. To obtain the cohomology of ck(Mg) itself, we consider the
vector bundle
)5( < : ~k(Mg)~km ~ k ck (Mg)+
which has the following properties. First, the Thom space of m times
~k is homomorphic to Dk(M ;sm). Secondly, it has finite even order,
g
see [CCKN]. Hence
(6) Dk (Mg;S2nk) = i2nk "k k C (Mg) +
for 2n k =ord(~). Thus we have
9
Theorem C. As a vector space, H~ck(Mg) ss__i isomorphic to the vector
subspace generated b~ all monomials of weight k in H~(g,nk , )
desuspended 2nkk times.
Regarding the homology of E =El w,z I .... ,Zig] we have
Theorem D. The homology H (E,d) is as follows:
)7(
rank Hi(2n+1)= -\i-2/ for i:O,I .... g, and all (non-zero)
elements have weight i;
{2gh {2g
(8) rank Hi(2n+1)+4n+£ - / i \ \i+2] for i =g ..... 2g, and all
(non-zero) elements have weight i+2;
)9( rank H. :0 in all other deqrees .j
3
Note the apparent duality rank Hj =rank HN_ j for = N 2g(2n+1)+4n+1.
We will give the proof of Theorem A in the next section. The proof
of Theorem B is the same as for [ BCT, Thm.B]. By what we said above
Theorem C folows from Theorem B. And Theorem D will be derived in the
last section.
3. Mapping spaces and fibrations. Let D denote an embedded disc in Mg.
There is a commutative diagram
(Io) C (D;S 2n) >~2s2n+2
C(M $;$2n) $ ;s2n+2)
>map o (Mg
(Mg, C ;$D s2n) > (&2n+2) 2g
where maPo stands for based maps. The right column is induced by
restricting to the l-section, and is a fibration. The left column is
a quasifibration. Since S 2n is connected, all three horizontal maps
01
are equivalences, see [M], [B] for details.
The E2-term of the Serre spectral sequence of these (quasi)fibrations
is as follows. From the base we have 2g-fold tensor product of
(11) H ~QS2n+2 : H~(s2n+IxQs 4n+3) =E[ zi] ~P[ul] (i : I,...2g),
where Izil = 2n+I and lui[ : 4n+2. From the fibre we have
(12) He~2S 2n+2 : H~(~s2n+Ix ~2s4n+3) : He(~S 2n+I x S 4n+I)
: [P v] m E[w] ,
where Ivl = 2n and lwl = 4n+I. The following determines all differentials
in this spectral sequence.
Lemma. The differentials are as follows:
(13) d2n+1(v) : O
(14) d4n+2(w) : 2ZlZ 2 + 2z2z 3 + ... + 2Z2g_iZ2g
Proof: Assertion (13) follows from the stable splitting of C(Mg;S2n) ,
on [ B]. (14) results from symmetries of M and of the fibrations (10)
g
which leave d invariant. •
S 2n)
The lemma implies E4n+3 :E = HwC(Mg; . Furthermore, E4n+3 is
a tensor product of the polynomial algebra P[v,u I .... U2g] and the
homology module H (E,d) of the exterior algebra E = E[w,z , I ...,Z2g]
with differential d. This proves Theorem A.
4. Homology of E. Let us write i = x z2i_1 and Yi = z2i for i = 1,...g.
The form d(w) = 2ZlZ 2 + 2z2z 3 + ... + 2Z2g,Z2g is equivalent to the
standard symplectie form xlY I +x2Y 2 + ... +Xgyg. The vector space