202
11 Discrete Morse Theory
condition that the matched cells form a regular pair (in the CW sense) is replaced by requiring that the covering weights in matched pairs be invertible.
However, the notion of acyclic matching, which is purely combinatorial,
11.3 Algebraic Morse Theory
205
and the last case is impossible by induction, and again, by the construction
of L.
), for all j, k. Indeed, this is clear for
We have w(bkj akj ) = w(bjk1 ak1
j
j = k. The case j < k follows by induction, and the case j > k
13.3 Further Facts About Nonevasiveness
241
Case 1: the primary case.
There exists an element y such that y is an atom of P and x y.
Proof of Claim 1.
Since y is an atom, L = L \ cfw_y is a lattice. Also, it is clear that x L . Set
B = B L . Then L \ B =
16.4 Spectral Sequences and Nerves of Acyclic Categories
0
1
q
x y
0
1 /21
0
0
0
0
1
2
283
p
Fig. 16.6. The E3, -tableau.
that respects filtration, i.e., for two filtered CW complexes
! =X
!t X
!t1 X
!0 ,
X = Xt Xt1 X0 and X
! such that (Xi ) X
!i , for
15.5 Gluing Spaces
271
(5) The conditions of the generalized nerve lemma are still satisfied, i.e., if the
!i s is nonempty, then it has to be (kt+1)-connected.
intersection of t of the A
Proof. We have the following verification:
"
#
!i A
!i = X
! Vi1 Vi
12.2 Lexicographic Shellability
217
t
2
1
q
y
1
r
1
p
1
1
c=x<p<q<y<t
d=x<r<y<t
x
Fig. 12.1. What can go wrong in the nonpure case.
only in the interval [x, y] such that c d, but c|[x,y] d|[x,y] . See Figure 12.1
for an example.
12.2.2 EL-Labeling
The fol
13.1 Evasiveness
229
already mentioned group homomorphism c. A general fact from field theory
tells us now that the group /H is cyclic.
Since the graph property G is nonevasive, we know that the abstract
simplicial complex (G) is collapsible; see Proposit
11.2 Discrete Morse Theory for CW Complexes
193
Example 1: The independence complexes of strings and cycles
Our first real application is concerned with the independence complexes of
graphs, which were defined in Subsection 9.1.1. Recall that for an arbit
13.2 Closure Operators
235
Application 2.
Let G be an arbitrary graph. Recall from the Subsection 9.1.4 that we
have an order-reversing map N : F(N (G) F(N (G), which maps every
set of vertices to their common neighbors. Recall also that the complex
(N (F
214
12 Lexicographic Shellability
12.1.2 Shelling Induced Subcomplexes
The next theorem provides a handy criterion for being able to conclude that
an induced subcomplex (see Definition 2.40) of a shellable complex is shellable
as well.
Theorem 12.5. Assum
196
11 Discrete Morse Theory
1|2|3|4
1|234
1|23|4
1|24|3
1|2|34
1|2|34
1|23|4
1|24|3
123|4
14|23
12|3|4
124|3
13|24
13|2|4
134|2
12|34
14|2|3
1|234
Fig. 11.10. The map for n = 4.
By Theorem 11.10 it is now sucient to construct acyclic matchings on
the fib
11.1 Discrete Morse Theory for Posets
181
acyclic. The reader is invited to intuitively think about such pairs as internal
collapses. The idea is to remove all the matched elements in some appropriate
order, so that the homotopy type of the underlying spa
11.2 Discrete Morse Theory for CW Complexes
187
Proof. The role of the base space here is played by the poset Q, and the
fiber maps gq are given by the acyclic matchings on the subposets 1 (q).
The decomposition theorem tells us that there exists a poset
300
17 Chromatic Numbers and the Kneser Conjecture
of V (G) with k weight(f ) independent sets, so that each element of V (G)
belongs to at least k of these independent sets.
Summarizing, we can conclude that if we limit our consideration to the
"f (G). S
14.1 Quotients of Nerves of Acyclic Categories
247
maps are the same as the original ones. Hence the quotient X/G is a generalized simplicial complex; see Definition 2.41.
"
Let us next consider a similar, yet somewhat dierent, situation, where X
is a fi
178
10 Acyclic Categories
!
g(x) = ! m=x f ( m), for all x O(C);
f (x) = m=x (m)g( m), for all x O(C).
Proof. The statement of the theorem is simply the translation of the fact
that f = g is equivalent to f = g. The latter is true since = e, using
(10.25)
12
Lexicographic Shellability
Lexicographic shellability is an important tool for studying the topological
properties of the order complexes of partially ordered sets. Although, as we
shall see in Remark 12.4, discrete Morse theory is more powerful as a m
10.6 The M
obius Function
!
( m) = 0.
175
(10.20)
m: m=x
In analogy with Halls theorem we have the following statement.
Theorem 10.26. For any finite acyclic category C with an initial object s
and a terminal object t, we have
(s) =
"(C).
(10.21)
Proof.
12.2 Lexicographic Shellability
223
Importantly, one has to know only how to compare labels on the edges on the
same level. In fact, if additionally, all edges leaving a vertex upward happen
to have dierent labels, then we need to know only how to compare
232
13 Evasiveness and Closure Operators
Fig. 13.2. A counterexample to the Generalized AanderaaRosenberg Conjecture.
conventions are used: the rectangular box with straight corners with one or
more numbers in it means ask whether these numbers are in our
226
13 Evasiveness and Closure Operators
pairs (i, j), for 1 i < j n, which correspond to potential edges in the graph.
A set of such pairs forms a simplex in (G) if and only if the corresponding
graph has the property G.
A graph property is called trivia
19.2 Examples of StiefelWhitney Test Graphs
f (m, n) = mf (m 1, n 1) + (m 1)f (m, n 1),
333
(19.3)
for n > m 2, with the boundary values f (n, n) = n! 1, f (1, n) = 0 for
n 1, and f (m, n) = 0 for m > n.
Then the generating function Fm (x) is given by the
15.3 Deforming Homotopy Colimits
265
(3) For the diagram from Example 15.9(3), we see that pb is the canonical
projection of the mapping cone onto an interval, whereas pf collapses the
actual cone inside the mapping cone to a point, which is the same as t
268
15 Homotopy Colimits
We cite the following theorem without proof.
Theorem 15.18. (Existence of a partition of unity)
If X is paracompact, and U is an open covering of X, then there exists a partition of unity that is subordinate to U. Moreover, if U i
330
19 Characteristic Classes and Chromatic Numbers
in Rn . Its dual, Mn , is the polytope associated to the hyperplane arrangement
A = cfw_A1 , . . . , An+1 defined by
!
(xi = 0),
for 1 i n;
Ai = "n
( j=1 xj = 0), for i = n + 1.
Let us identify each cel
14.3 Conditions on Group Actions
253
For language convenience, if one of Conditions (C1) and (C2) is satisfied,
we shall also say that Condition (C) is satisfied. In this case, the map is
an isomorphism.
g
m1
g
m2
g
mt1
ma
mb
g
Fig. 14.4. Condition (C1).
318
18 Structural Theory of Morphism Complexes
Proposition 18.10. . For any fixed graph K, the Hom construction yields
a contravariant functor Hom (, K).
Proof. We only need to check the following fact: for any sequence of
graph homomorphisms : T G and :
262
15 Homotopy Colimits
Again, the topological spaces and continuous maps can be replaced with
any other category. In fact, when C is a category, a diagram of topological
spaces over C is nothing but a functor D : C Top.
We note that when considering a p
306
17 Chromatic Numbers and the Kneser Conjecture
index being (S, [n]\S). Though we already know at this point that the cubical
complex KCn,k is homotopy equivalent to a wedge of (n k)-dimensional
spheres, we would now like to describe an acyclic matchin