20.3 Encoding Cohomology Generators by Arc Pictures
357
Since lists over vertices in V are already maximized, we can only add an element to one of the single-element lists over vertices in S \ V . Each such list
assignment is obtained in exactly two ways:
1 Overture
3
As another example, we refer to a combinatorial computation of some
concrete Stiefel-Whitney characteristic classes in Theorem 19.13 in Subsection 19.2.2. There, once the combinatorial description of the characteristic
classes has been found,
354
20 Applications of Spectral Sequences to Hom Complexes
Proof. By Corollary 20.8, we just need to check that
nk 1 > m + n 4,
(20.1)
where k = (m + 1)/3. If m cfw_5, 6, then k = 2, and (20.1) reduces to
n + 3 > m, which is true. Thus we can assume that
382
[dL03]
References
M. de Longueville, 25 Jahre Beweis der Kneservermutung der Beginn
der topologischen Kombinatorik (German), [25th anniversary of the proof
of the Kneser conjecture. The start of topological combinatorics], Mitt.
Deutsch. Math.-Ver. 20
388
Index
link of a simplex, 11
locally trivial fiber bundle, 111
long exact sequence of a pair, 83
Lov
asz complex, 134
Lov
asz conjecture, 337
Lov
asz test, 303
M
obius function for acyclic categories,
174
Main theorem of discrete Morse theory
for CW co
Preface
IX
Acknowledgments
Many organizations as well as individuals have made it possible for this book
project to be completed. To start with, it certainly would not have materialized without the generous financial support of the Swiss National Science
20.6 Cohomology with Integer Coecients
375
Assume first that m = 3k 1. We see from Table 20.1 that the top (in
mk,(n2)(k1)
terms of the sum of coordinates) nonzero element is E2
. This
element is on the diagonal m k + (n 2)(k 1) = m + nk 3k n + 2 =
nk n +
372
20 Applications of Spectral Sequences to Hom Complexes
It is now crucial to realize that the same collapses can still be performed in
the cochain complex (At , d1 ), even though we are working over the integers.
This follows from the version of discre
20.1 Hom+ Construction
353
2
1
1,3
2
1,2,3
2
1
3
1,2,3
Hom+ (K2 , )
2
[2]
Fig. 20.2. The support map from Hom+ (K2 , ) to [2] .
crucial at this point, since an infinite simplex does not have a well-defined
barycenter.
An alternative concise way to phrase
2.1 Abstract Simplicial Complexes
15
removed tuples are indexed by (
i1 , . . . ,
il1 , k ), satisfying the condition
i1 il1 k .
After that, the new simplices are added. By the definition of the stellar
subdivision together with our induction assumption,
20.5 Euler Characteristic Formula
367
Proposition 20.21. The nonzero entries E2p,q , for p = m 1, are the folmt2,t(n2)
mt1,t(n2)
= E2
= Z2 , for 2 t
lowing: E2m3,n2 = Z2 , E2
!
2m/31,m(n2)/3
0,0
!
= Z32 .
(m1)/3, and E2 = Z2 . If additionally 3 m, then E
368
20 Applications of Spectral Sequences to Hom Complexes
Theorem 20.23. For an arbitrary graph T , we have the following formula:
"
!(Hom (T, Kn ) =
(1)n+|S|
!(Ind(T [S])n .
(20.12)
=SV (T )
Proof. By Corollary 20.6 we know that Hom+ (T, Kn ) = Ind (T
356
20 Applications of Spectral Sequences to Hom Complexes
Let H q (Hom (T [S], G); Z2 ), for some q and some S V (G). The value of
the first dierential on is then given by
!
iG [S cfw_x, S]().
(20.6)
dp,q
1 () =
xV (T )\S
We shall next detail the formula
374
20 Applications of Spectral Sequences to Hom Complexes
20.6.4 Summary: the Full Description of the Groups
! (Hom (Cm , Kn ); Z)
H
For an abelian group , we let (d) denote the copy of in a graded Zalgebra, placed in dimension d.
Theorem 20.27. For any
Preface
The intent of this book is to introduce the reader to the beautiful world of
Combinatorial Algebraic Topology. While the main purpose is to describe the
modern research tools and latest applications of this field, an attempt has
been made to keep
2.2 Polyhedral Complexes
23
ai2 ai4 ai1 ai3 ai5 ai6
Fig. 2.2. The horizontal levels correspond to the successive steps of the algorithm
computing the value of the function f .
One way to think about the barycentric subdivision, which can come in
handy in
20.2 Setting up the Spectral Sequence
355
20.2.2 The 0th and the 1st Tableaux
Next, we shall describe the 0th and the 1st tableaux of this spectral sequence,
and then perform a partial analysis of the second tableau.
To start with, as an additional piece
2.1 Abstract Simplicial Complexes
9
Definition 2.6. Let 1 and 2 be two finite abstract simplicial complexes.
A simplicial map from 1 to 2 is a set map f : V (1 ) V (2 ) such that
if is a simplex of 1 , then f () is a simplex of 2 . In such a situation, we
1
Overture
The subject of Combinatorial Algebraic Topology is in a certain sense a classical one, since modern algebraic topology derives its roots from dealing with
various combinatorially defined complexes and with combinatorial operations
on them. Yet
366
20 Applications of Spectral Sequences to Hom Complexes
Proposition 20.20. For any natural numbers m, n, and g such that n >
m(g1), the complex Thinm,n,g consists of (m+n)/ gcd(m, n) cubes, each one
having dimension gcd(m, n), which are connected toget
References
[Kem79]
[KLS93]
[Kn55]
[Ko97]
[Ko98]
[Ko99]
[Ko00]
[Ko01]
[Ko02]
[Ko05a]
[Ko05b]
[Ko05c]
[Ko06a]
[Ko06b]
[Ko06c]
[Ko06d]
[Ko07]
[Kr92]
[Kr00]
381
A.B. Kempe, On the geographical problem of four colors, Amer. J. Math.
2, (1879), pp. 193204.
S. K
20.6 Cohomology with Integer Coecients
369
Let T and G be two graphs, and let us fix an order of the vertices of T
and of G. When convenient, we may identify vertices of T , resp. of G, with
integers 1, . . . , |V (T )|, resp. 1, . . . , |V (G)|, accordin
370
20 Applications of Spectral Sequences to Hom Complexes
Proof. We need to check that the map commutes with the dierentials,
, where supp = V (T ). Comand it is enough to do this for a generator +
paring the incidence number from (20.15) for Hom (T, G)
20.6 Cohomology with Integer Coecients
373
Case 1. 2 x m 1. By formula (20.19), filling in x 1 or x will have
the same sign. Furthermore, when filling in x, we shall also need to move the
vertex representative from x + 1 to x. These vertices have dierent