Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Stably Free Modules over the Klein Bottle
Andrew Misseldine
Brigham Young University
3 April 2011
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Polynomials
Definition - Skew Laurent Polynomials
Let R be a ring and let σ : R → R be a ring isomorphism.
A skew Laurent polynomial is a polynomial which allows negative
exponents for the indeterminate x and with the relation that
xr = σ(r)x, ∀r ∈ R.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Polynomials
Definition - Skew Laurent Polynomials
Let R be a ring and let σ : R → R be a ring isomorphism.
A skew Laurent polynomial is a polynomial which allows negative
exponents for the indeterminate x and with the relation that
xr = σ(r)x, ∀r ∈ R.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Polynomials
Definition - Skew Laurent Polynomials
Let R be a ring and let σ : R → R be a ring isomorphism.
A skew Laurent polynomial is a polynomial which allows negative
exponents for the indeterminate x and with the relation that
xr = σ(r)x, ∀r ∈ R.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Polynomials
Definition - Skew Laurent Polynomials
Let R be a ring and let σ : R → R be a ring isomorphism.
A skew Laurent polynomial is a polynomial which allows negative
exponents for the indeterminate x and with the relation that
xr = σ(r)x, ∀r ∈ R.
Then the collection of all skew Laurent polynomials with
coefficients from R is a ring.
Denote this ring as
R[x, x −1 ; σ].
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Projective Modules
Definition - Stably Free Module
An R-module P is stably free iff there exists natural numbers m, n
such that P ⊕ R m ∼ = R n .
{Free Modules} ⊆ {Stably Free Modules} ⊆ {Projective Modules}
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Projective Modules
Definition - Stably Free Module
An R-module P is stably free iff there exists natural numbers m, n
such that P ⊕ R m ∼ = R n .
{Free Modules} ⊆ {Stably Free Modules} ⊆ {Projective Modules}
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
This is a Klein Bottle
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
This is a Klein Bottle
x ❄
•
v0
✲ y
✛
y
X
x
❄
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
This is a Klein Bottle
x ❄
•
v0
✲ y
✛
y
We see from the fundamental square, that
Call this group G.
X
x
❄
π(X , v0) = 〈x, y | x −1 yx = y −1 〉
= 〈x, y | yx = xy −1 〉
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
This is a Klein Bottle
x ❄
•
v0
✲ y
✛
y
We see from the fundamental square, that
Call this group G.
X
x
❄
π(X , v0) = 〈x, y | x −1 yx = y −1 〉
= 〈x, y | yx = xy −1 〉
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
This is a Klein Bottle
x ❄
•
v0
✲ y
✛
y
We see from the fundamental square, that
Call this group G.
X
x
❄
π(X , v0) = 〈x, y | x −1 yx = y −1 〉
= 〈x, y | yx = xy −1 〉
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
This is a Klein Bottle
x ❄
•
v0
✲ y
✛
y
We see from the fundamental square, that
Call this group G.
X
x
❄
π(X , v0) = 〈x, y | x −1 yx = y −1 〉
= 〈x, y | yx = xy −1 〉
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
This is a Klein Bottle
x ❄
•
v0
✲ y
✛
y
We see from the fundamental square, that
Call this group G.
X
x
❄
π(X , v0) = 〈x, y | x −1 yx = y −1 〉
= 〈x, y | yx = xy −1 〉
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The Ring ZG
In light of the presentation of G,
ZG = Z[y, y −1 ][x, x −1 ; σ]
and σ is the isomorphism induced by
σ : y ↦−→ y −1 .
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The Ring ZG
In light of the presentation of G,
ZG = Z[y, y −1 ][x, x −1 ; σ]
and σ is the isomorphism induced by
In particular,
as polynomials in Z[y, y −1 ].
σ : y ↦−→ y −1 .
yn = ny, ∀n ∈ Z
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The Ring ZG
In light of the presentation of G,
ZG = Z[y, y −1 ][x, x −1 ; σ]
and σ is the isomorphism induced by
In particular,
as polynomials in ZG.
σ : y ↦−→ y −1 .
yn = ny, ∀n ∈ Z
xn = nx, ∀n ∈ Z
yx = xy −1
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The Ring ZG
In light of the presentation of G,
ZG = Z[y, y −1 ][x, x −1 ; σ]
and σ is the isomorphism induced by
In particular,
as polynomials in ZG.
σ : y ↦−→ y −1 .
yn = ny, ∀n ∈ Z
xn = nx, ∀n ∈ Z
yx = xy −1
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The ZG-module K
Let
r = 1 + y + y 3 ∈ ZG,
s = 1 + y −1 + y −3 ∈ ZG.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The ZG-module K
Let
Let
r = 1 + y + y 3 ∈ ZG,
s = 1 + y −1 + y −3 ∈ ZG.
K = (r) ∩ (x + s)
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The ZG-module K
Let
Let
r = 1 + y + y 3 ∈ ZG,
s = 1 + y −1 + y −3 ∈ ZG.
K = (r) ∩ (x + s)
= rx 2 − sr 2 , rs(x + s)
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The ZG-module K
Let
Let
r = 1 + y + y 3 ∈ ZG,
s = 1 + y −1 + y −3 ∈ ZG.
K = (r) ∩ (x + s)
= rx 2 − sr 2 , rs(x + s)
Let π : ZG ⊕ ZG → ZG as
f1
π = rf1 + (x + s)f2.
f2
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
K is Stably Free
Theorem (Exactness)
The sequence 0 −→ K −→ ZG ⊕ ZG π
−−−→ ZG −→ 0 is exact,
that is, the map π : ZG 2 → ZG is surjective and ker π ∼ = K.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
K is Stably Free
Theorem (Exactness)
The sequence 0 −→ K −→ ZG ⊕ ZG π
−−−→ ZG −→ 0 is exact,
that is, the map π : ZG 2 → ZG is surjective and ker π ∼ = K.
Thus, ZG 2 K ∼ = ZG. Since ZG is free, the sequence splits and we
have that
K ⊕ ZG ∼ = ZG 2 .
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
K is Stably Free
Theorem (Exactness)
The sequence 0 −→ K −→ ZG ⊕ ZG π
−−−→ ZG −→ 0 is exact,
that is, the map π : ZG 2 → ZG is surjective and ker π ∼ = K.
Thus, ZG 2 K ∼ = ZG. Since ZG is free, the sequence splits and we
have that
K ⊕ ZG ∼ = ZG 2 .
Corollary
K is a stably free ZG-module.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
K is Stably Free
Theorem (Exactness)
The sequence 0 −→ K −→ ZG ⊕ ZG π
−−−→ ZG −→ 0 is exact,
that is, the map π : ZG 2 → ZG is surjective and ker π ∼ = K.
Define : ZG → ZG ⊕ ZG as
=
sx −2
x −1 − rx −2
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
K is Stably Free
Theorem (Exactness)
The sequence 0 −→ K −→ ZG ⊕ ZG π
−−−→ ZG −→ 0 is exact,
that is, the map π : ZG 2 → ZG is surjective and ker π ∼ = K.
Define : ZG → ZG ⊕ ZG as
=
We note that
sx −2
x −1 − rx −2
π = 1ZG
and is a splitting of the short exact sequence.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Presentation of K
0 −→ ZG
−−−→ ZG ⊕ ZG −→ K −→ 0
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Presentation of K
0 −→ ZG
−−−→ ZG ⊕ ZG −→ K −→ 0
K ∼ = ZG 2 / im
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Presentation of K
0 −→ ZG
−−−→ ZG ⊕ ZG −→ K −→ 0
K ∼ = ZG 2 / im
K = 〈e1, e2 | (1)〉
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Presentation of K
0 −→ ZG
−−−→ ZG ⊕ ZG −→ K −→ 0
K ∼ = ZG 2 / im
K = 〈e1, e2 | (1)〉
K = 〈e1, e2 | e1s + e2(x − r)〉 .
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
K is Not Free
Theorem (Stafford (1985))
Let R be a commutative Noetherian domain. Suppose
S = R[x, x −1 ; σ] is a skew Laurent extension of R with elements
r, s ∈ R such that
1 r is not a unit in S,
2 rS + (x + s)S = S,
3 σ(r)s /∈ rR.
Then the S-module K = rS ∩ (x + s)S is a non-free, stably free
right ideal of S, satisfying K ⊕ S ∼ = S 2 .
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Infinitely Many Stably Free ZG-modules
Let
with n = 0 and
Define
rn = 1 + ny + ny 3
sn = 1 + ny −1 + ny −3 .
Kn = (rn) ∩ (x + sn) .
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Infinitely Many Stably Free ZG-modules
Let
with n = 0 and
Define
rn = 1 + ny + ny 3
sn = 1 + ny −1 + ny −3 .
Kn = (rn) ∩ (x + sn) .
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Infinitely Many Stably Free ZG-modules
Let
with n = 0 and
Define
Corollary
rn = 1 + ny + ny 3
sn = 1 + ny −1 + ny −3 .
Kn = (rn) ∩ (x + sn) .
Kn is a non-free, stably free right ideal of ZG for all non-zero
integers n.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Infinitely Many Stably Free ZG-modules
How do we know if
for n = m?
Kn ∼ = Km
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Infinitely Many Stably Free ZG-modules
How do we know if
for n = m?
Theorem (Artamonov (1981))
Kn ∼ = Km
There exists countably many nonisomorphic non-free, stably free
ZG-modules. In particular, ∃Q ⊆ N such that ∀p, q ∈ Q, Kp ∼ = Kq
and Q is infinite.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
(G, 2)-Complexes
Definition
Let G be a group. A (G, 2)-complex is a 2-dimensional
CW-complex with fundamental group G.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
(G, 2)-Complexes
Definition
Let G be a group. A (G, 2)-complex is a 2-dimensional
CW-complex with fundamental group G.
Let X be the CW-complex of the Klein Bottle and let G = π0(X ).
Thus, X is a (G, 2)-complex.
x ❄
•
v0
✲ y
✛ y
A
X
x
❄
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Euler Characteristic
We would like to classify all homotopy types of (G, 2)-complexes
with a fixed Euler Characteristic n.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Euler Characteristic
We would like to classify all homotopy types of (G, 2)-complexes
with a fixed Euler Characteristic n.
Theorem (Asphericity)
The Klein bottle, with χ(X ) = 0, has the minimal Euler
characteristic of all (G, 2)-complexes. Furthermore, the Klein
Bottle is the only (G, 2)-complex with Euler characteristic equal to
0, up to homotopy.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Euler Characteristic
We would like to classify all homotopy types of (G, 2)-complexes
with a fixed Euler Characteristic n.
Theorem (Asphericity)
The Klein bottle, with χ(X ) = 0, has the minimal Euler
characteristic of all (G, 2)-complexes. Furthermore, the Klein
Bottle is the only (G, 2)-complex with Euler characteristic equal to
0, up to homotopy.
We use Algebra to next classify (G, 2)-complexes with Euler
characteristic 1.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Algebraic (G, 2)-complexes
Definition
An algebraic (G, 2)-complex is an exact sequence
C∗ : F2 → F1 → F0 → Z → 0
where the Fi are finitely generated, free ZG-modules.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Algebraic Complexes of G
The Klein Bottle X has the cellular chain complex
C∗(X ) : Z〈A〉 −→ Z〈x, y〉 −→ Z〈v0〉 −→ 0.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Algebraic Complexes of G
The Klein Bottle X has the cellular chain complex
Then,
C∗(X ) : Z〈A〉 −→ Z〈x, y〉 −→ Z〈v0〉 −→ 0.
C∗( X ) : ZG ∂2
∂1 ε
−→ ZG ⊕ ZG −→ ZG −→ Z −→ 0
is an algebraic (G, 2)-complex.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Algebraic Complexes of G
The Klein Bottle X has the cellular chain complex
Then,
C∗(X ) : Z〈A〉 −→ Z〈x, y〉 −→ Z〈v0〉 −→ 0.
C∗( X ) : ZG ∂2
∂1 ε
−→ ZG ⊕ ZG −→ ZG −→ Z −→ 0
is an algebraic (G, 2)-complex.
Notice that
H2( X ) = ker(∂2) = 0.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Algebraic Complexes of G
Next, let
X1 = X ∨ S 2 .
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Algebraic Complexes of G
Next, let
X1 = X ∨ S 2 .
So, X1 has the cellular chain complex
C∗(X1) : Z〈A, S 2 〉 −→ Z〈x, y〉 −→ Z〈v〉 → 0
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Algebraic Complexes of G
Next, let
X1 = X ∨ S 2 .
So, X1 has the cellular chain complex
and
C∗(X1) : Z〈A, S 2 〉 −→ Z〈x, y〉 −→ Z〈v〉 → 0
C∗( X1) : ZG ⊕ ZG ∂2⊕0
−−−−−→ ZG ⊕ ZG ∂1
−→ ZG ε
−→ Z −→ 0.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Algebraic Complexes of G
Next, let
X1 = X ∨ S 2 .
So, X1 has the cellular chain complex
and
C∗(X1) : Z〈A, S 2 〉 −→ Z〈x, y〉 −→ Z〈v〉 → 0
C∗( X1) : ZG ⊕ ZG ∂2⊕0
−−−−−→ ZG ⊕ ZG ∂1
−→ ZG ε
−→ Z −→ 0.
Notice that
H2( X1) = ker(∂2 ⊕ 0) = ZG.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The Algebraic Complex K∗
We now will construct an algebraic (G, 2)-complex with no obvious
geometric interpretation. Let K∗ be the exact sequence
K∗ : ZG ⊕ ZG ∂2◦π
−−−−−→ ZG ⊕ ZG ∂1
−→ ZG ε
−→ Z −→ 0.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The Algebraic Complex K∗
We now will construct an algebraic (G, 2)-complex with no obvious
geometric interpretation. Let K∗ be the exact sequence
K∗ : ZG ⊕ ZG ∂2◦π
−−−−−→ ZG ⊕ ZG ∂1
−→ ZG ε
−→ Z −→ 0.
But, H2(K∗) = K ∼ = ZG = H2(C∗(X1)).
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The Algebraic Complex K∗
We now will construct an algebraic (G, 2)-complex with no obvious
geometric interpretation. Let K∗ be the exact sequence
K∗ : ZG ⊕ ZG ∂2◦π
−−−−−→ ZG ⊕ ZG ∂1
−→ ZG ε
−→ Z −→ 0.
But, H2(K∗) = K ∼ = ZG = H2(C∗(X1)).
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
The Algebraic Complex K∗
We now will construct an algebraic (G, 2)-complex with no obvious
geometric interpretation. Let K∗ be the exact sequence
K∗ : ZG ⊕ ZG ∂2◦π
−−−−−→ ZG ⊕ ZG ∂1
−→ ZG ε
−→ Z −→ 0.
But, H2(K∗) = K ∼ = ZG = H2(C∗(X1)).
Therefore,
K∗ C∗(X1).
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Geometric Realization
Theorem
There exist chain-homotopically distinct, algebraic
(G, 2)-complexes with Euler characteristic 1, where G is the
fundamental group of the Klein bottle.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Geometric Realization
Theorem
There exist countably many chain-homotopically distinct,
algebraic (G, 2)-complexes with Euler characteristic 1, where G is
the fundamental group of the Klein bottle.
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
Geometric Realization
Theorem
There exist countably many chain-homotopically distinct,
algebraic (G, 2)-complexes with Euler characteristic 1, where G is
the fundamental group of the Klein bottle.
Question
Does K∗ arise as an algebraic complex of some geometric
(G, 2)-complex?
Andrew Misseldine Stably Free Modules over the Klein Bottle

Introduction
Constructing a Stably Free Module
Applications to the Klein Bottle
References
Harlander, Jens and Andrew Misseldine, ”K-Theoretic and
Homotopic Aspects of the Klein Bottle Group” (2011).
Preprint.
Misseldine, Andrew, “Stably Free Modules Over the Klein
Bottle” (2010). Boise State University Theses and
Dissertations. Paper 106.
http://scholarworks.boisestate.edu/td/106
Andrew Misseldine Stably Free Modules over the Klein Bottle