Smith Theory Revisited
William G. Dwyer
Clarence W. Wilkerson*
1 Introduction
In the late 1930's, P. A. Smith began the investigation of the cohomological
properties of a group G of prime order p acting by homeomorphisms on a topo-
logical space X. This thread has continued now for almost fifty years. Smith was
successful in calculating the cohomology of the the fixed point sets for involu*
*tions
on spheres [Smith 1] and projective spaces [Smith 2]. In the 1950's, Smith theo*
*ry
was reformulated by the introduction of the Borel construction XG = EG xG X
and equivariant cohomology H*(XG ) = H*G(X). Borel made the key observa-
tion [Borel 1] that the cohomology of the fixed point set was closely related to
the torsion-free (with respect to H*G(pt)) quotient of H*G(X) . In the 1960's,
this was formalized as the "localization theorem" of Borel-Atiyah-Segal-Quillen
[Atiyah-Segal],[Quillen]. (The localization theorem is described below.) The lo-
calization theorem has previously been used to deduce the actual cohomology of
the fixed point set for particular examples in an ad hoc fashion, but a general
algorithmic computation of the cohomology of the fixed point set has not been
provided in the literature.
______________________________
*The research of the authors was partially supported by the NSF, and that of*
* the
second author by sabbatical funds from Wayne State University.
1
The main result of this note is that the localization theorem provides such a
description of the unlocalized equivariant cohomology of the fixed point set al*
*so.
One simply has to use the data given by the equivariant cohomology as a module
over both H*G(pt) and the mod p Steenrod algebra, Ap.
2 Statement of Results
The theory is valid for G an elementary abelian p-group, and X a finite G-CW
complex. By default, cohomology is understood to have coefficients in the field
Fp of p elements. For K a subgroup of G, define S = S(K) to be the multi-
plicative subset of H*(BG) generated by the Bockstein images in H2(BG) of
the elements x in H1(BG) which restrict non-trivially to H1(BK):
The general form of the localization theorem appearing in [Hsiang, Chapter
] specializes the following statement.
2.1_Theorem_(Localization Theorem) [ ]) The localized restriction map
S-1H*G(X) -! S-1H*G(XK )
is an isomorphism.
For geometric reasons both H*G(X) and H*G(XK ) are unstable Ap-modules.
By [Wilkerson] the localization S-1H*G(X) and S-1H*G(XK ) inherit Ap-module
structures themselves; these induced structures, however, do not satisfy the in*
*sta-
bility condition. For any graded Ap-module M let Un (M) denote the subset
of "unstable classes", i.e., Un (M) is the graded Fp-vector space defined as
follows:
(1) if p = 2
Un (Mk = {x 2 Mk | Sqi(x) = 0; i > k}
(2) if p is odd
Un (M)2k = {x 2 M2k | Pi(x) = 0 (i > k); fiPi(x) = 0 (i k)}
Un (M)2k+1 = {x 2 M2k+1 | Pi(x) = 0 (i > k); fiPi(x) = 0 (i > k)}
2
It is true [Adams-Wilkerson] (although nontrivial) that Un (M) is closed under
the Ap-action on M and is thus an unstable Ap-module. The localization map
H*G(XK ) -! S-1H*G(XK )
respects the Ap-action and therefore lifts to a map
H*G(XK ) -! Un (S-1H*G(XK )):
2.2_Theorem_ The above map
H*G(XK ) -! Un (S-1H*G(XK )):
is an isomorphism.
2.3_Remark_ The objects in Theorem 2.2 are algebras, Ap-modules and modules
over H*(BG); the isomorphism respects these additional structures.
The equivariant cohomology H*G(XG ) is just the tensor product H*(BG)Fp
H*(XG ); so the special case K = G of Theorem 2.2 leads to the following corol-
lary.
2.4_Corollary_There is a natural isomorphism
H*(XG ) Fp H*(BG) Un (S(G)-1H*G(X)):
2.5_Remark_Corollary 2.4 gives a direct functorial calculation of the cohomo*
*l-
ogy of the fixed-point set XG in terms of the Borel cohomology H*G(X): It was
the question of whether such a calculation was possible which led to the present
work. Note that by [Lannes 1,2] there is a result parallel to 2.4 which express*
*es
H*(XG ) in terms of H*G(X) and the functor "T" of Lannes. We intend to pursue
some further connections between 2.4 and the work of Lannes in a subsequent
paper.
3 Some algebraic preliminaries
We will assume here and in the next section that p is odd; the arguments need
straight forward indexing changes to work in the case p = 2. Let oe be the cycl*
*ic
3
group of order p and let R denote the graded commutative ring H*(Boe): Recall
that R is the tensor product over Fp of an exterior algebra on an element a 2 R1
with a polynomial algebra on the Bockstein element b 2 fi(a): The ring R is
also an unstable Ap-module in such a way that multiplication in R satisfies the
Cartan formula; given b = fi(a); the action of Ap is determined by the equation
P1(b) = bp:
3.1_Definition_An unstable Ap-R module is a non-negatively graded Fp-vector
space M which is both a graded R-module and a graded Ap-module in such a
way that the multiplication map R Fp M -! M satisfies the Cartan formula.
3.2_Example_ If N is an unstable Ap-module, then the Cartan formula itself
gives the natural structure of an unstable Ap - R module to R Fp N:
If M is an unstable Ap - R module, then for x 2 Mk define (x) by the
formula
[k=2]X
(x) = (-b)i(p-1)P[k=2]-i(x)
i=0
3.3_Lemma__ The operation has the following properties
(i) (x + y) = (x) + (y)
(ii) (bx) = 0
(
a(x) | x | even
(iii) (ax) = p-1
b a(x) | x | odd
(iv) if x=b 2 Un (b-1M); then bk(x) = 0 for some k 0:
3.4_Remark_ In part (iv) of 3.3, the implicit Ap-module structure on b-1M is
as always the one provided by [Wilkerson].
Proof_of_3.3_Part(i) is clear, while (ii) and (iii) follow from the Cartan form*
*ula.
For (iv), observe that if x 2 Mk then the image of (x) in b-1M is exactly
bP[k=2](x=b); this must vanish in b-1M if x=b is an unstable class.
If N is an Fp-vectorPspace and x is a non-zero element of R Fp N; write
x as a (finite!) sum 1i=0xi with xi 2 R Fp N and define the leading_term_of
x to be the unique xi such that xj = 0 forj > i:
4
3.5_Lemma__ Suppose that N is an Fp-vector space and that x1; . .;.xj are
non-zero elements of R Fp N such that the leading term of each xi can be
written as bki mi; where the k0is are non-negative integers and the m0is are
elements of N which are linearly independent over Fp: Then the elements
x1; . .;.xj are linearly independent overPR, i.e., given a collection r1; . .;.*
*rj
of elements of R, the linear combination ji=1rixi vanishes in R Fp N iff
each coefficient ri is zero.
P j
Proof_It is necessary only to inspect the leading term of i=1rixi: The lemma
depends only on the fact that b is not a zero-divisor in R:
3.6_Proposition_If N is an unstable Ap-module, then the natural map
R Fp N -! Un (b-1(R Fp N))
is an isomorphism.
Proof_It is clear that b does not annihilate any non-zero element of R Fp N:
By 3.3(iv), then, it is sufficient to show that given x 2 R Fp N with (x) = 0
there exists y 2 R Fp N such that x = by: Assume x 6= 0; and write
Xj
x = a"ibki mi
i=1
where " 2 {0; 1}; ki 0 and the elements m1; . .;.mj are linearly independent
over Fp: By 3.3 (ii)-(iii)
Xj
(x) = ri(mi)
i=1
where ri = 0 iffki > 0; i = 1; . .;.j: However, by 3.5 and the definition of
the elements (m1); . .;.(mj) are linearly independent over R: Consequently,
the vanishing of (x) implies that each ri is zero which entails that each ki is
greater than zero and thus that x is divisible by b inR Fp N:
4 Completion of the Proof
We will continue to use the notation of x3. A two-dimensional element of S(K)
will be called a generator; by definition, every element in S(K) is a product of
5
generators.
4.1_Lemma__For each generator y of S(K) there exists a retraction f : G ! oe
and an unstable Ap-module N such that
(i) f*(b) = y ; and
(ii) H*G(XK ) with the R-action induced by f* is isomorphic as an unstable
Ap - R module to R Fp N:
Proof_ Write y = fi(x) forx 2 H1(BG) and letf : G -! oe be the unique
surjection such that f*(a) = x: By the definition of S(K) it is possible to find
a section i : oe - ! G of f such that i(oe) K: Use f and i to identify G
with the product oe x J where J = ker(f); since oe acts trivially on XK via th*
*is
identification, the K"unneth formula produces an isomorphism
H*G(XK ) H*oexJ(XK ) H*(Boe) Fp H*J(XK )
The lemma follows with N = H*J(XK ):
Proof_of_2.2_By 4.1 no generator of S(K) annihilates any non-zero element of
H*G(XK ); this implies that the map
H*G(XK ) -! Un (S(K)-1H*G(XK ))
is injective. To prove surjectivity, note that each element of S(K)-1H*G(XK )
can be written as x=s withx 2 H*G(XK ) and s 2 S(K): It is clear that if x=s is
an unstable class, then so is t . (x=s) fort 2 S(K): Therefore, by induction on
the length of an expression for s as a product of generators, it is enough to s*
*how
that if x belongs to H*G(XK ) and y is a generator of S(K); then x=y is unstable
iff x=y belongs to H*G(XK ): This follows immediately from 4.1 and 3.6.
5 Involutions on Cohomology Real Projec-
tive Spaces
The classification of the fixed point sets for involutions on cohomology real p*
*ro-
jective spaces provides an interesting illustration of the previous theory. For*
* actual
real projective spaces, the cohomological classification was done by [Smith 2],
6
and the generalization to cohomology projective spaces appears in Su, Bredon
- see also the books of Bredon [Bredon 1] and W.Y. Hsiang [Hsiang]. There
are versions for higher rank elementary p-groups and complex and quarterionic
projective spaces, but the classical case should suffice to demonstrate the pos*
*si-
bilities of our description.
The guiding principle is that on the cohomological level, the general case m*
*im-
ics the special case of involutions of real projective space RP ninduced by lin*
*ear
involutions on euclidean space Rn+1. For such an involution, the eigenspaces for
{ +1, -1 } in Rn+1 correspond to the two components of the fixed point set
of the action on RP n. Each component is a real projective space of dimension
less than or equal to n. On the cohomology level, the equivariant cohomology
H*G(RP n) is the cohomology of the RP n-bundle over BG associated to the
vector bundle arising from the representation of G on Rn+1 . The Leray-Hirsh
theorem gives this explicitly as the free H*(BG)-module on { xi; 0 i n }
with the multiplicative relation that xi(x + w)j = 0; where w is the generator *
*of
H*(BG); i is the dimension of the {+1}-eigenspace, and j is the dimension of
the {-1}-eigenspace.
In Proposition 5.1 below we show that in the general case, the possibilities*
* for
the equivariant cohomology rings all coincide with those already present in this
special case. The fact that the equivariant cohomology determines completely
the cohomology of the fixed point set ( by Corollary 2.4 ), then concludes the
proof. That is, in the general case the cohomology of the fixed point is that of
a disjoint union of two real projective spaces, or is empty.
However, for the benefit of algebraically inclined readers, we also calculate
directly in Proposition 5.2 the cohomology of the fixed point set from the lo-
calized equivariant cohomology ring. Analogues of both Propositions 5.1 and
5.2 could be carried out for actions of elementary 2-groups with no essentially
new information. For complex and quaterionic projective cases, more possibiliti*
*es
for the Steenrod algebra action occur, but otherwise, the reasoning applies in a
similar fashion.
Proposition 5.1 : If G = Z=Z2Z and X is a finite G - CW -complex with
H*(X) = F2[x]=xn+1 , then H*G(X) = F2[w; y]=(f) , where y restricts to x and
7
f = yi(y + w)j for i + j = n + 1:
Proposition 5.2 : If R = H*G(X), then in S-1R,
ffl 2ffl
1) for any ffl max(i; j) the elements = (x=w)2 and = ((x + w)=w) are
idempotent and hence unstable.
2) The elements {xs; 0 s < i; (x + w)t; 0 t < j } are a basis for
S-1R over F2[w; w-1], and a basis for Un(S-1R) over F2[w].
3) The F2-subalgebra generated by (x) is isomorphic to H*(RP i-1) and
similarly for that generated by (x + w).
Proof of Proposition 5.1 : The Serre spectral sequence for the fibration X !
XG ! BG collapses if there is a fixed point, and hence H*G(X) is a free mod-
ule over H*(BG) on generators {1; y; . .;.yn } , with an multiplicative relation
yn+1 = g(x; y) where g has degree less than n + 1 in y: Since w and y are each
in dimension one, the natural surjection from F2[w0; y0] onto R is an A2 map.
Hence g = (y0)n+1 + f(w0; y0) generates a principal ideal in F2[w0; y0] which is
closed under the A2- action. By Serre [Serre], or Wilkerson [Wilkerson], such a
polynomial is a product of linear combinations of w0and y0. However, since R is
free over F2[w], the term w can not occur in the product. In this two-dimension*
*al
case, the only other possibilites are y and (y + w).
em Proof of Proposition 5.2 : The fact that and are idempotents follows
directly from the observations
ffl 2ffl
(y=w)2 + ((y + w)=w) = 1
and ffl
((y=w)(y + w)=w)2 = 0
because of the choice of large ffl. Since these elements have grading zero and *
*are
idempotent, the Cartan formula demonstrates that the elements are unstable.
8
Clearly, xi = 0 and (x + w)j = 0, so x and (x + w) do generate alge-
bras isomorphic to those for projective spaces. The union of these two algebras
consists of (n + 1) linearly independent elements over F2[w; w-1], so they form
a basis for S-1R . If M is this union, then in fact
S-1R = M F2 F2[w; w-1]
as A2 - H*(BG)- modules, since both M and F2[W; W -1] are closed under
the A2 action. Hence the unstable elements in S-1R are just M F2 F2[w] by
Proposition 3.6.
References
[Adams-Wilkerson] Adams, J.F., Wilkerson, C.W., Finite H-spaces and
Algebras over the Steenrod Algebra, Annals of Math.
(111) 1980, 95-143.
[Atiyah-Segal] Atiyah, M.F., Segal, G., Equivariant cohomology and
localization, lecture notes, 1965, Warwick.
[Borel 1] Borel.A. et. al.,Seminar on transformation groups,
Annals of Math. Studies 46, Princeton (1960).
[Borel 2] Borel, Armand, Nouvelle demonstration d'un theorem
de P.A. Smith, Comment. Math. Helv. 29 27-39
(1955).
[Bredon 1] Bredon, Glen
E.,Introduction_to_Compact_Transformation_Groups_.
Academic Press (New York) 1972.
[Bredon 2] Bredon, Glen E., The cohomology ring structure of a
fixed point set, Annals of Math 80, 524-537 (1964).
[Hsiang] Hsiang, W.Y.,
Cohomological_Theory_of_Transformation_Groups_,
Springer Verlag (New York), 1975.
9
[Lannes 1] Lannes, J., Sur la cohomologie modulo p des p-groups
abeliens elementaires, to appear.
[Lannes 2] Lannes, J., Cohomology of groups and function
spaces, to appear.
[Quillen] Quillen, D., The spectrum of an equivalent cohomol-
ogy ring I, II, Annals Math. 94 (1971), 549-572, 573-
602.
[Serre] Serre, J.-P.,Sur la dimension cohomologique des
groupes profinis, Topology 3 (1965) 413-420.
[Smith 1] Smith, P.A., Transformations of finite period, Annals
of Math 39 (1938), 127-164.
[Smith 2] Smith, P.A., New results and old problems in finite
transformation groups, Bull. Amer. Math Soc. 66, 401-
415 (1960).
[Su] Su, J.C., Transformation groups on cohomology pro-
jective spaces, Trans. Amer. Math Soc. 106, 305-318
(1964).
[Wilkerson] Wilkerson, C.W., Classifying Spaces, Steenrod Opera-
tions and Algebraic Closure, Topology (16) 1977, 227-
237.
10