COHOMOLOGY WITH
&
Sunil Chebolu
Illinois State University
ISMAA Annual meeting,
Bradley University, April 3-4 2009.
Plan of my talk
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


Motivation
Cohomology
Cohomology with stones and sticks
Applications
1. Groups acting on spheres
2. Generating hypothesis
Plan of my talk




Motivation
Cohomology
Cohomology with stones and sticks
Applications
1. Groups acting on spheres
2. Generating hypothesis
Motivation
k - Field of characteristic p (p > 0 prime)
G - Finite group
kG - Group ring
= {  g 2 G αg g | αg 2 k }
Ultimate Goal: Explore the hidden secrets of finite
groups.
A traditional approach: One studies finite groups
by examining how they act on k - vector spaces.
A simple example
C2 = h ¾ | ¾2 = 1 i acts on k2 = k © k
¾ (v) = -v
¾2(v) = ¾ (¾ (v))
V
= ¾(-v)
= -(-v)
-V
=v
Therefore ¾2 = 1
This is a 2-dimensional representation of C2
Definition: An n-dimensional representation
of G is a homomorphism of groups
½ : G ! GLn(k).
GLn(k) = group of all invertible n £ n
matrices over k.
Example: C2 ! GL2(k)
¾  – I2
1  I2
Thus, an n-dimensional representation of G
is an n-dimensional vector space V over k
on which G acts by linear automorphisms.
||
n-dimensional kG - module
Fundamental problem: Given G, classify all
of its representations.
This problem is notoriously hard !
To study finite groups and their representations,
a very powerful algebraic machine was introduced
in the early 20th century.
This is Group Cohomology --- one of the biggest
inventions in the 20th century.
G finite group
+
RG cohomology ring
+, x
The cohomology ring RG is a very useful object to
study invariants of G
Henri Poincaré
David Hilbert
The history of cohomology is long and rich
– Early 1900s: Inception
– Late 1940s: Matured
– Thereafter: active area of research
Three avatars of cohomology
– Representation theory (G = finite group)
Group cohomology
– Topology (G = topological group)
Continuous cohomology
– Number theory (G = Galois group)
Galois cohomology
Plan of my talk




Motivation
Cohomology
Cohomology with stones and sticks
Applications
1. Groups acting on spheres
2. Generating hypothesis
Cohomology
RG = H*(G, k) = ©n ≥ 0 Hn(G,k) is a ring.
char(k) divides |G|  Modular case
Towards the definition of cohomology:
kG linear map: This is a map Á : M ! N between
representations M and N
– Á is a linear transformation
– Á (g ¢ m) = g Á(m) for all g in G and m in M.
HomG(M, N) = all kG-linear maps from M to N.
Examples of representations
– Trivial representation of G is k
with trivial G action, i.e., gx = x
– Regular representation of G is kG
kG = {  g 2 G αg g | αg 2 k }
– Free representation of G is
kG © kG ©  © kG
– Sygizies of a representation M are
1 M = ker( F1 ³ M)
2 M = ker( F2 ³ 1 M)
n M = ker( Fn ³ n-1 M) for n > 1.
Definition of Hn(G,M)
Hn(G,M) = HomG(n k, M)/~
f,g : n k ! M are homotopic (f~g) iff
f – g: n k
M
F
H*(G, M) = ©n ≥ 0 Hn(G,M)  graded k-vector
space.
Notation: Hom(A, B) = HomG(A, B)/~
Important special case: M = k (trivial rep.)
H*(G, k) = ©n ≥ 0 Hn(G,k)  graded ring
Ring structure:
® 2 H2(G, k)
) ®¯ 2 H5(G,k)
¯ 2 H3(G, k)
¯
®
2
3
 k
k
 k
®¯ :
5
k
3®
3
k =
3
k
¯
k
k
Plan of my talk




Motivation
Cohomology
Cohomology with stones and sticks
Applications
1. Groups acting on spheres
2. Generating hypothesis
Cohomology with stones and sticks
C2 = h ¾ | ¾2 = 1 i
Char(k) = 2
Freshman’s dream
kC2 = k[¾]/(¾2 – 1) = k[¾]/(¾ -1)2 = k[x]/(x2)
k=
kC2 =
x
 k = ker( kC2 ! k)
= ker
)
=
i k = k
for all i.
= k
Hi(C2, k) = Hom(i k, k) = Hom(k, k) = kh xii
H*(C2, k) = ©i≥0 kh xi i
x1 x1 = x12 2 H2(C2,k) = h x2 i
Products:
1
k
x2 : k
x1
x1
1
k
x1
k
k = k
x1
x 12 = x 2
Similarly, x1n = xn.
)
H*(C2, k)  k[x1]
k
k
Klein four group V4
V4 = C2 © C2
= h ¾, ¿ | ¾2 = 1 = ¿2 = 1, ¾ ¿ = ¿ ¾ i
This is our favourite group !
kV4 = k[x, y]/(x2, y2)
= kh1, x, y, xy i
k=
x = ¾ -1, y = ¿ -1
kV4 =
x
y
y
x
Sygizies of k
1 k = ker (kV4 ³ k)
= ker
=
2 k = ker (kV4 © kV4 ³ 1 k)
= ker
©
=
Similarly, we have
n k =

n copies
-n k =

n copies
Computing H*(V4, k)
H1(V4, k) = Hom(1 k, k)
= Hom
=k
,
u
=khui©khvi
©k
v
H2(V4, k) = Hom(2 k, k)
= Hom
,
=k
l
©
k
m
©
k
r
=khli©khmi©khri
©
Products
Educated guess: l = u2, m = uv, r = v2 !
u2 = l
u
u
u.u =
u
=
u2
=
uv = m
u.v =
v
u
v
=
=
uv
v2 = r
v
v
vv=
v
=
=
v2
Hn(V4, k) = Hom(n k, k)
= Hom

,
= k-span h un, un-1v, , uvn-1, vn i .
Combining all these,we have:
H*(V4, k) = © Hn(V4, k)
= © k-span h un, un-1v, , uvn-1, vn i
= k [u, v]
Plan of my talk




Motivation
Cohomology
Cohomology with stones and sticks
Applications
1. Groups acting on spheres
2. Generating hypothesis
Applications
§1. Groups acting freely on spheres.
Problem: Given a finite group G, can it act
freely on some sphere (Sn, for some n).
Definition: An action Á : G £ Sn ! Sn is free
if Á(g, x) = x ) g = e .
Theorem: If G acts freely on Sn then
H*(G,k) is periodic with period dividing n.
Special case: G = V4
Recall: H*(V4, k)  k[u, v].
In particular, Hn(V4, k) is a k-vector space of
dimension n+1 with basis { un, un-1v, , vn }.
) limn ! 1 dimk Hn(V4, k) = limn ! 1 n+1 = 1
Conclusion: V4 cannot act freely on any sphere.
!!! YAY !!!
§2. Generating hypothesis
-- joint work with Carlson and Mináč.
Jan Mináč
Jon Carlson
Induced map in cohomology
G finite group and M, N kG –modules
Á : M ! N be a kG-linear map.
Á induces a map in cohomology:
Hi(G, Á) : Hi(G, M) ! Hi(G, N)
®Á±®
籨:
i
k
®
M
Á
N
The problem of generating hypothesis
Suppose
Hi(G, Á) : Hi(G, M) ! Hi(G, N)
is 0 for all i>0.
Does this imply that Á : M ! N is null-homotopic?
i.e., does Á factor through a free representation?
Remark: The converse of the above problem
is true for a trivial reason.
M
Á
F
N
) Hi(G, Á) = 0 because
Hi(G, F) = 0 8 i > 0
Theorem: (Carlson, C, Mináč)
Generating hypothesis holds for G
The p-Sylow subgroup of G = C2 or C3
A map h of kV4 modules
h is not null
homotopic
h is zero in cohomology
Ars Longa Vita Brevis
Thank you