A random Bockstein operator
As more of topology’s tools become popular in analyzing high-dimensional data sets, the goal of understanding the underlying probabilistic properties of these tools becomes even more important. While much attention has been given to understanding the probabilistic properties of methods that use homo...
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| Zitieren: | A random Bockstein operator / M.J. Zabka // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 311–321. — Бібліогр.: 5 назв. — англ. |
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| citation_txt | A random Bockstein operator / M.J. Zabka // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 311–321. — Бібліогр.: 5 назв. — англ. |
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| description | As more of topology’s tools become popular in analyzing high-dimensional data sets, the goal of understanding the underlying probabilistic properties of these tools becomes even more important. While much attention has been given to understanding the probabilistic properties of methods that use homological groups in topological data analysis, the probabilistic properties of methods that employ cohomology operations remain unstudied. In this paper, we investigate the Bockstein operator with randomness in a strictly algebraic setting.
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“adm-n2” — 2018/7/24 — 22:32 — page 311 — #149
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 25 (2018). Number 2, pp. 311–321
c© Journal “Algebra and Discrete Mathematics”
A random Bockstein operator
Matthew Zabka
Communicated by V. Lyubashenko
Abstract. As more of topology’s tools become popular in
analyzing high-dimensional data sets, the goal of understanding the
underlying probabilistic properties of these tools becomes even more
important. While much attention has been given to understanding
the probabilistic properties of methods that use homological groups
in topological data analysis, the probabilistic properties of methods
that employ cohomology operations remain unstudied. In this paper,
we investigate the Bockstein operator with randomness in a strictly
algebraic setting.
1. Introduction
Using the tools of algebraic topology to better understand a data set is a
relatively new idea with many applications. For example, Carlsson’s survey
[1] reviews the generalization of cluster analysis to persistent homology, a
technique that provides more information on the shape of a data set than
traditional cluster analysis. Other authors, such as Kahle in [3], have
investigated the topology of a random simplicial complex.
Both of these approaches have only considered Betti numbers, i.e., the
ranks of cohomology groups. A natural question that arises is whether
one can gather any additional information from a data set by looking at
operations on the topological structure generated by that data set. That
is, how can we expand the idea of randomness to cohomology operations?
The Bockstein homomorphism is a well-known example of a cohomology
2010 MSC: 08, 55.
Key words and phrases: random cohomology operations, topological data anal-
ysis, Bockstein operation.
“adm-n2” — 2018/7/24 — 22:32 — page 312 — #150
312 A random Bockstein operator
operator, and in this paper, we shall attempt to investigate this cohomology
operator with randomness.
Cohomology operators are a topological invariants that can reveal
additional structure not seen in cohomological groups. For example, the
cohomology groups of for S1 ∨ S2 and RP
2 are the same, but they are not
homotopy equivalent spaces. To see this, one can compute the Bockstein
homomorphism of both S1 ∨ S2, which is trivial, and of RP2, which is
non-trivial.
In general, the Bockstein homomorphism is a connecting homomor-
phism of cohomology groups defined on a chain complex. Ideally, we should
consider the case of a chain complex of a randomly generated topological
space. Unfortunately, this problem is very difficult. The length of the
chain complex, each Abelian group in the complex, and each boundary
map would all add complexity to this model. We shall therefore examine
in this paper a simpler algebraic version of the above problem whose only
degrees of freedom are determined by a single boundary map.
Let V and W be free-modules with coefficients in Z/p2. We have then
have the following short exact sequences
0 → pV →֒ V ։ V → 0 and 0 → pW →֒W ։W → 0,
where V and W are the reductions of V and W mod p. Given a map
φ : V → W , which is the boundary map we describe in the paragraph
above, define ψ from V to W to be the map induced by φ. The Bockstein
homomorphism induced by φ is then a map from kerψ to cokerψ. We
give construction of the Bockstein homomorphism for this case in more
detail in Section 4.
Since Bockstein homomorphisms are elements of hom(kerψ, cokerψ),
it makes sense only to compare Bocksteins induced by functions from V to
W that are equal modulo p. If V has dimension n and W has dimension
m, then a choice of random function from V to W is the same as choosing
a random m by n matrix. To this end, let φ be a random matrix whose
entries are chosen i.i.d. randomly from the discrete uniform distribution
on {0, 1, 2, . . . , p2− 1}. Let φ be the reduction of φ modulo p. Let ψ be an
m by n matrix with entries in {0, 1, 2, . . . , p− 1}. Let βφ the be Bockstein
homomorphism induced by φ. Let β be in hom(kerψ, cokerψ). We shall
show that
P
(
βφ = β
∣
∣φ = ψ
)
=
1
pk(m−n+k)
.
In other words, we shall show that, conditioned on φ = ψ, the Bockstein
homomorphisms are distributed uniformly.
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M. Zabka 313
2. Linear algebra over ZZZ/p2
Many of our calculations will be done over Z/p2-modules. This section
reviews the theory of Z/p2-modules over Z/p2. Some of the techniques
used in this section work for modules over rings other than Z/p2, but we
shall not explore these ideas here.
Let R be a ring. Given an R-module M , we say that a subset E of M
is a basis for M whenever E generates M and E is linearly independent.
This definition is equivalent to the condition that every x in M can be
written as a unique linear combination of elements of E with scalars in R.
A module that has a basis is called a free module.
Let p be prime, and let V and W be free Z/p2-modules. Define
V := V
⊗
Z/p2
Z/p and W :=W
⊗
Z/p2
Z/p.
So V = V/pV andW =W/pW are the reductions of V andW mod p.
Note that these are Z/p vector spaces. For an element x ∈ V , we use x to
denote its reduction modulo p. For an element y in V , we use ỹ to denote
a choice of representative in V of y, so that ỹ = y. Given a Z/p2-linear
map φ : V →W , let φ denote the induced function from V to W .
Lemma 2.1. Let V be a free Z/p2-module. Let p : V → V be multipli-
cation by p. Then the kernel of p is equal to the image of p.
Proof. Let {ei} be a basis for V . Let x be in ker p. Since {ei} is a basis,
there are αi in Z/p2 such that x =
∑
i αiei. Since x is in ker p we have
px =
∑
i pαi · ei = 0. By the independence of the ei, we have pαi = 0 for
each i. Thus αi = pβi for some βi ∈ Z/p2. Then p(
∑
i βiei) =
∑
i αiei = x.
So that x is in the image of p.
Next, assume that y is in the image of p. Then there exists a z ∈ V
with pz = y. So py = p2z = 0. So y is in the kernel of p.
We know that pV and V are isomorphic as Z/p-vector spaces, because
they both have the same dimension. The following lemma gives an explicit
isomorphism between these two spaces.
Lemma 2.2. The map f : pV → V defined by px 7→ x is a Z/p-linear
isomorphism.
Proof. We show that both f and its inverse mapping g, which maps x in
V to px in pV , are well-defined. To show that f is well-defined, assume
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314 A random Bockstein operator
that px = py for some x and y in V . Then px − py = p(x − y) = 0. So
x− y = pz for some z ∈ V by Lemma 2.1. Note that
x− y = x− y = pz = 0,
so that f is well-defined.
For the inverse mapping g, suppose x = y. Then x− y = 0. So by
Lemma 2.1, x− y = pz for some z ∈ V . We have
px− py = p(x− y) = p2z = 0.
So g is well-defined. By inspection we see that both f and g are Z/p-linear
functions, and so the proof is complete.
The main proposition of this section shows that any lift of a basis of
V is a basis of V . Such bases will be useful for constructing linear maps
out of V . That is, if one defines a map on any basis of V , then this map
extends linearly to all of V .
Proposition 2.3. Let {ei} be a basis for V . For each ei, let ẽi in V be
any lift of ei. Then {ẽi} is a basis for V .
Proof. We first show that the set {ẽi} is linearly independent. Suppose
αi ∈ Z/p2 with
∑
i
αiẽi = 0. (1)
Projecting to V we obtain
∑
i αiei = 0. Since {ei} is a basis for V , we
must have that αi = 0 for every i. So each αi = pβi for some βi in Z/p2.
Thus, (1) gives that
∑
i βi · pẽi = 0 in pV . Under the isomorphism given
in Lemma 2.2, we have
∑
i βiei = 0 in V . Since the set {ei} is linearly
independent, each βi = 0, so each βi = pγi for some γi in Z/p2. This gives
that each αi = pβi = p2γi = 0. So the set {ẽi} is linearly independent.
We next show that {ẽi} spans V . Let x ∈ V . Since the set {ei} is
a basis for V , there are αi ∈ Z/p2 such that
∑
i αiei = x. So for some
y ∈ V ,
x = py +
∑
i
αiẽi. (2)
Under the isomorphism given in Lemma 2.2, the element py in pV
is mapped to y in V . Since the ei form a basis for V , there exist βi in
“adm-n2” — 2018/7/24 — 22:32 — page 315 — #153
M. Zabka 315
Z/p2 such that
∑
i βiei = y. Thus pz +
∑
i βiẽi = y for some z ∈ V .
Substituting this into (2) gives
x = p
(
pz +
∑
βiẽi
)
+
∑
i
αiẽi.
Simplifying gives x =
∑
i(αi − pβi)ẽi, so that x is in the span of {ẽi}, as
desired.
For the map ψ with domain V and target W , recall that cokerψ is
defined as the quotient W/ψ(V ). Our next lemma shows that we may
regard the Bockstein homomorphism as a map β : kerψ → cokerψ.
The techniques used in the proof are similar to the techniques used in
Lemma 2.2.
Lemma 2.4. The map f from pW/φ(pV ) to cokerψ defined by
f : pw + φ(pV ) → w + ψ(V )
is an isomorphism.
Proof. We must show that f and its inverse mapping g are well-defined.
To show that f is well-defined, suppose pw + φ(pV ) = pw′ + φ(pV )
in pW/φ(pV ). We must show that w − w′ is in ψ(V ). We have that
p(w − w′) ∈ φ(pV ). Thus p(w − w′) = pφ(v) for some v ∈ V . By Lemma
2.1, we have w − w′ − φ(v) = py. Thus w − w′ = φ(v) = ψ(v). So w − w′
is in ψ(V ), and this shows that f is well-defined.
We next want to show that the inverse mapping g is well-defined.
Suppose that w + ψ(V ) = w′ + ψ(V ). We must show that pw + φ(pV ) =
pw′ + φ(pV ).Since w − w′ + φ(V ) = 0 + φ(V ), there exists a v ∈ V with
w−w′ = φ(v). Thus w−w′−φ(v) = px for some x, which, by Lemma 2.1
gives p[w−w′− pφ(v)] = 0. So pw+φ(pV ) = pw′+φ(pV ). By inspection,
f and g are both linear, and the proof is complete.
3. Spaces of linear maps
We should like to further investigate the connection between a map
ψ : V → W and the Bockstein homomorphisms induced by a map
φ : V →W such that φ = ψ. For this section, we shall treat ψ as a fixed
Z/p-linear map from V to W .
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316 A random Bockstein operator
Definition 3.1. Let V and W be Z/p2-modules. Let V and W be the
reductions of V and W modulo p. Let ψ be a fixed Z/p-linear map from
V to W . Define Lψ to be the collection of all maps from V to W whose
reduction modulo p is ψ.
It will also be useful in this section to choose a basis for V , which, by
Proposition 2.3 will lift to a basis for V .
Definition 3.2. Let V , V , and ψ be as in Definition 3.1. Let {ei} ∪ {fj}
be a basis for V such that {ei} is a basis for the subspace kerψ of V . For
each i, let ẽi in V be a lift of ei. For each j let f̃j in V be a lift of fj .
By Proposition 2.3, {ẽi}∪{f̃j} is a basis for V . If the map ψ : V →W
is not the zero map, then we know that Lψ is not a vector space, for in
this case, 0 is not in Lψ. This fact, along with the next lemma, gives that
Lψ is a vector space if and only if ψ is the zero map.
Lemma 3.3. The space L0 with pointwise addition and Z/p scalar mul-
tiplication defined by
α · φ := α · φ,
where α is in Z/p2 and φ is in L0, is a Z/p-vector space. In particular,
if V has dimension n and W has dimension m, then L0 is a Z/p-vector
space of dimension m · n.
Proof. We shall only show that this scalar multiplication is well-defined,
as the other parts of the proof are straightforward. Let α1 and α2 be in
Z/p2 with α1 = α2. Let φ be in L0 and let v ∈ V . Then α1 − α2 = pβ for
some β in Z/p2 and φ(v) = pw for some w in W , because φ = 0. So we
have
α1 · φ(v)− α2 · φ(v) = α1φ(v)− α2φ(v)
= (α1 − α2)φ(v)
= (pβ)(pw)
= p2βw
= 0,
which shows that this scalar multiplication in Z/p2 is well-defined.
Let φ0 be any element of Lψ. Then φ0+L0 = Lψ, so we may regard Lψ
as a coset of L0. It will be useful however to choose a particular φ0 ∈ Lψ
whenever we wish to regard Lψ as a coset of L0. For this, we need only
define φ0 on the basis {ẽi} ∪ {f̃j} given in Definition 3.2.
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M. Zabka 317
Remark 3.4. When we regard Lψ as a coset of L0, we shall choose φ0
such that, for all i, φ0(ẽi) = 0, and for all j, φ0(f̃j) is any value whose
reduction modulo p is ψ(fj).
We are now ready to count the number of elements in Lψ.
Lemma 3.5. For any ψ : V →W , the set Lψ has pmn elements.
Proof. Lemma 3.3 tells us that L0 is a Z/p-vector space, but by definition,
L0 also is a Z/p2-submodule of Hom(V,W ). When we regard Lψ as
φ0 + Lψ, where φ0 is as defined in Remark 3.4, this addition occurs in a
Z/p2-submodule. So, while Lψ is not a translate of L0 as a Z/p-vector
space, we still know that Lψ has the same number of elements as L0. This
information, along with Lemma 3.3, completes the proof.
4. The Bockstein homomorphism
What follows is a short review of the Bockstein homomorphism in
the context that is relevant for our study of cohomology operations with
randomness. Several references cover the Bockstein homomorphism and
cohomology operations in more generality. See, for example, [4].
As in Section 2, let V and W be Z/p2 free-modules with coefficients
in Z/p2. We have the following short exact sequences:
0 → pV →֒ V ։ V → 0 and 0 → pW →֒W ։W → 0,
where V and W are the reductions of V and W mod p.
Consider a Z/p2-linear map φ from V to W . Let ψ be the map from
V to W induced by φ. Then the Snake Lemma [5] defines a map β with
domain kerψ and target pW/φ(pV ). The following diagram illustrates
the Snake Lemma.
kerψ
pV V V
pW W W
pW/φ(pV )
β
φ ψ
“adm-n2” — 2018/7/24 — 22:32 — page 318 — #156
318 A random Bockstein operator
More precisely, for v ∈ kerψ, choose any representative v ∈ V of v.
Since the squares in the above diagram commute, we have φ(v) = ψ(v) = 0.
So φ(v) = pw for some w ∈ W . Define the Bockstein homomorphism β
from kerψ to pW/φ(pV ) by
β(v) := pw + φ(pV ).
The following diagram shows the process described above.
v v
pw pw = φ(v)
pw + φ(pV )
φ
By construction, the target of β is pW/φ(pV ). However, by Lemma 2.4,
we know that pW/φ(pV ) is isomorphic to cokerψ. So henceforth we shall
regard β as a map into cokerψ.
Remark 4.1. We note here that if one regards an arbitrary chain complex,
the map β is often called a connecting homomorphism. When the chain
complex is generated by a topological space, the map β is called the
Bockstein homomorphism. If we regard φ as the map between V and W
in the following chain complex
. . . 0 V W 0 . . . ,
φ
and consider the reduced chain complex
. . . 0 V W 0 . . . ,
ψ
then the only possible non-trivial homology groups of this chain complex
are kerψ and cokerψ. Although we are in a strictly algebraic setting, we
shall continue to refer to the map β as the Bockstein homomorphism
between kerψ and cokerψ.
Remark 4.2. The Bockstein homomorphism is often constructed in the
case where V and W are Z-modules. In this case, first reduce V and W
to Z/p2 modules, and then apply the above construction.
“adm-n2” — 2018/7/24 — 22:32 — page 319 — #157
M. Zabka 319
In this section we have described how to every φ ∈ Lψ, there is a
unique Bockstein homomorphism βφ : kerψ → cokerψ. This fact defines
the following map.
Definition 4.3. Define Γ to be the map from Lψ to Hom(kerψ, cokerψ)
that sends φ in Lψ to the unique Bockstein homomorphism βφ, which is
in Hom(kerψ, cokerψ), that is given by φ.
Composing Γ with addition by φ0 gives a well defined set map B with
domain L0 and target Hom(kerψ, cokerψ). This is shown in the following
diagram.
L0 Lψ Hom(kerψ, cokerψ)
+φ0
B
Γ
We should like to examine the properties of this map. The map from
L0 to Lψ given by adding φ0 is a bijection. The next lemma shows that
the map Γ is onto, which shows that B is also onto. In particular, every
Z/p linear map from the kernel of ψ to the cokernel of ψ is the Bockstein
homomorphism of some φ : V →W that induces ψ.
Lemma 4.4. The map Γ from Lψ to Hom(kerψ, cokerψ) is onto.
Proof. Let β ∈ Hom(kerψ, cokerψ). Let {ei} ∪ {fj} and {ẽi} ∪ {f̃j} be
bases of V and V as defined in Definition 3.2. We shall define φ on the
basis for V and then extend linearly to define φ on all of V . We must
show that the Bockstein homomorphism βφ of φ is equal to β.
For each i, we know that ei is in the domain of β. So β(ei) = wi+ψ(V )
for some wi in W . Define φ(ẽi) = pwi. Define φ(f̃j) to be any value in W
whose reduction modulo p is ψ(fj). Then φ(ei) = pwi = 0 = ψ(ei) and
φ(f̃j) = ψ(fj). This shows that φ = ψ. In particular φ is in Lψ.
By construction, βφ(ei) = wi + ψ(V ) = β(ei). Since βφ is equal to β
on the basis of kerψ, they are equal as Z/p-linear functions.
Lemma 4.5. The map B is a Z/p-linear map.
Proof. Let φ and φ′ be in L0. We must show that Γ(φ + φ′ + φ0) =
Γ(φ + φ0) + Γ(φ′ + φ0), for φ0 ∈ Lψ as described in Remark 3.4. For a
basis {ei} of kerψ, it suffices to show that
Γ(φ+ φ′ + φ0)(ei) = Γ(φ+ φ0)(ei) + Γ(φ′ + π0)(ei)
“adm-n2” — 2018/7/24 — 22:32 — page 320 — #158
320 A random Bockstein operator
Let ẽi be any lift of ei. Then φ0(ẽi) = 0 by construction. Also,
(φ+ φ′)(ẽi) = φ(ẽi) + φ′(ẽi) = 0, (3)
and
φ(ẽi) = φ′(ẽi) = 0, (4)
because φ and φ′ are in L0. Thus there are wi and w′
i in W with φ(ẽi) = pw
and φ′(ẽi) = pw′. Thus by (4) we know that
Γ(φ+ φ0)(ei) + Γ(φ′ + φ0)(ei) =
(
wi + ψ(V )
)
+
(
w′
i + ψ(V )
)
= wi + w′
i + ψ(V ).
Equations (3) and (4) together give that
Γ(φ+ φ′ + φ0)(ei) = wi + w′
i + ψ(V ).
So Γ(φ+ φ′ + φ0)(ei) = Γ(φ+ φ0)(ei) + Γ(φ′ + φ0)(ei), as desired.
5. Counting
Let V and W be Z/p2-modules of dimensions n and m respectively.
In Section 3 we defined Lψ as the collection of all maps from V to W
whose reduction modulo p is ψ. We then found that Lψ has pm·n elements.
We should next like to answer the following question: Given a Bockstein
homomorphism β, which is in hom(kerψ cokerψ), how many φ in Lψ have
β as their Bockstein homomorphisms? To answer this question, we shall
first look at the size of Γ−1(β).
Lemma 5.1. Let k := dim(kerψ). Then the space Γ−1(β) has exactly
p(m+k)(n−k) elements.
Proof. Since translation by the φ0 given in Remark 3.4 is a bijection, we
know that B−1(β) has the same size as Γ−1(β). Since B is a linear map,
we also know that B−1(0) has the same size as B−1(β). Thus Γ−1(β) has
the same size as B−1(0), so we shall find the size of B−1(0).
Since V and W have dimension n and m respectively, we know by
Proposition 2.3 that V and W also have dimensions n and m respectively.
Recall that cokerψ is defined as W/ψ(V ). Let k := ker(ψ). Since ψ
is a Z/p-linear map, by the Rank-Nullity Theorem, we have that n =
k+dim(ψ(V )). So dim(ψ(V )) = n−k. Thus dim(coker(ψ)) = m−(n−k).
From this, we have that the number of elements in B−1(0) is
pmn−k(m−n+k) = p(m+k)(n−k).
“adm-n2” — 2018/7/24 — 22:32 — page 321 — #159
M. Zabka 321
We now come to our main result. Recall that a choice of random
function from V to W is the same as choosing a random m by n matrix.
Theorem 5.2. Let φ be a random m by n matrix whose entries are
chosen i.i.d. from the discrete uniform distribution on {0, 1, 2, . . . , p2 − 1}.
Let φ be the reduction of φ modulo p. Let ψ be a matrix with entries in
{0, 1, 2, . . . , p − 1}. Let βφ be the Bockstein homomorphism defined by
φ as in Definition 4.3. Note that βφ is a random variable. Let β be in
hom(kerψ, cokerψ). Then
P
(
βφ = β
∣
∣φ = ψ
)
=
1
pk(m−n+k)
.
Proof. We know from Remark 3.5 that Lψ has pmn elements. By Lemma
4.4, we know that Γ is onto, and by Lemma 5.1, we know that the size of
hom(kerψ, cokerψ) is pmn−k(m−n+k).
References
[1] G. Carlsson, Topology and data, Bulletin of the American Mathematical Society,
N.2, 2009, pp.255-308.
[2] M. Kahle, Topology of random clique complexes, Discrete Mathematics, N.6, 2009,
pp.1658-1671.
[3] M. Kahle, Random geometric complexes, Discrete & Computational Geometry, N.3,
2011, pp.553-573.
[4] Mosher, Robert E. and Tangora, Martin C., Cohomology Operations and Applications
in Homotopy Theory, Dover, 2008.
[5] Atiyah, Michael F. and Macdonald, Ian G., Introduction to Commutative Algebra,
Addison-Wesley Reading, 1969.
Contact information
Matthew Zabka Dept of Mathematics and Computer Science,
SM 178, Southwest Minnesota State University,
1501 State Street, Marshall, MN 56258, USA
E-Mail(s): matthew.zabka@smsu.edu
Received by the editors: 25.05.2016
and in final form 24.06.2017.
|
| id | nasplib_isofts_kiev_ua-123456789-188366 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-01T23:20:43Z |
| publishDate | 2018 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Zabka, M.J. 2023-02-25T15:10:08Z 2023-02-25T15:10:08Z 2018 A random Bockstein operator / M.J. Zabka // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 2. — С. 311–321. — Бібліогр.: 5 назв. — англ. 1726-3255 2010 MSC: 08, 55. https://nasplib.isofts.kiev.ua/handle/123456789/188366 As more of topology’s tools become popular in analyzing high-dimensional data sets, the goal of understanding the underlying probabilistic properties of these tools becomes even more important. While much attention has been given to understanding the probabilistic properties of methods that use homological groups in topological data analysis, the probabilistic properties of methods that employ cohomology operations remain unstudied. In this paper, we investigate the Bockstein operator with randomness in a strictly algebraic setting. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics A random Bockstein operator Article published earlier |
| spellingShingle | A random Bockstein operator Zabka, M.J. |
| title | A random Bockstein operator |
| title_full | A random Bockstein operator |
| title_fullStr | A random Bockstein operator |
| title_full_unstemmed | A random Bockstein operator |
| title_short | A random Bockstein operator |
| title_sort | random bockstein operator |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/188366 |
| work_keys_str_mv | AT zabkamj arandombocksteinoperator AT zabkamj randombocksteinoperator |