Download Open subgroups of free topological groups

Free topological groups
Topological π1
Semicoverings
Proof Sketch
Open subgroups of free topological groups
Jeremy Brazas
Spring Topology and Dynamics Conference, New Britain, CT
March 23, 2013
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Free topological groups
The free Graev topological group on a based space (X , e) is the
topological group FG (X , e) equipped with a based map σ : X → FG (X , e):
Given f : X → G, f (e) = eG
X
/ FG (X , e)
σ
f
# ∃!e
f
G
TopGrp(FG (X , e), G) Top∗ (X , U(G))
The unbased version is the free Markov topological group FM (X ).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Free topological groups
The free Graev topological group on a based space (X , e) is the
topological group FG (X , e) equipped with a based map σ : X → FG (X , e):
Given f : X → G, f (e) = eG
X
/ FG (X , e)
σ
f
# ∃!e
f
G
TopGrp(FG (X , e), G) Top∗ (X , U(G))
The unbased version is the free Markov topological group FM (X ).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Free topological groups
The free Graev topological group on a based space (X , e) is the
topological group FG (X , e) equipped with a based map σ : X → FG (X , e):
Given f : X → G, f (e) = eG
X
/ FG (X , e)
σ
f
# ∃!e
f
G
TopGrp(FG (X , e), G) Top∗ (X , U(G))
The unbased version is the free Markov topological group FM (X ).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Free topological groups
The free Graev topological group on a based space (X , e) is the
topological group FG (X , e) equipped with a based map σ : X → FG (X , e):
Given f : X → G, f (e) = eG
X
/ FG (X , e)
σ
f
# ∃!e
f
G
TopGrp(FG (X , e), G) Top∗ (X , U(G))
The unbased version is the free Markov topological group FM (X ).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Free topological groups
The free Graev topological group on a based space (X , e) is the
topological group FG (X , e) equipped with a based map σ : X → FG (X , e):
Given f : X → G, f (e) = eG
X
/ FG (X , e)
σ
f
# ∃!e
f
G
TopGrp(FG (X , e), G) Top∗ (X , U(G))
The unbased version is the free Markov topological group FM (X ).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Free topological groups
I
Introduced by
Markov (1940s) FM (X )
Graev (1960s) FG (X , e)
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Free topological groups
I
Introduced by
Markov (1940s) FM (X )
Graev (1960s) FG (X , e)
I
Free topological groups take the place of free groups in the
general study of topological groups.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Free topological groups
I
Introduced by
Markov (1940s) FM (X )
Graev (1960s) FG (X , e)
I
Free topological groups take the place of free groups in the
general study of topological groups.
I
Typically, X is Tychonoff (σ : X ⊂ FG (X , e)).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Free topological groups
I
Introduced by
Markov (1940s) FM (X )
Graev (1960s) FG (X , e)
I
Free topological groups take the place of free groups in the
general study of topological groups.
I
Typically, X is Tychonoff (σ : X ⊂ FG (X , e)).
I
As groups,
FM (X ) = F(X )
FG (X , e) = F(X \e) = F(X )/heiF(x)
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topological Nielsen-Schreier?
Nielsen-Schreier Theorem: Every subgroup of a free group is free.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topological Nielsen-Schreier?
Nielsen-Schreier Theorem: Every subgroup of a free group is free.
Question: Is every subgroup H ≤ FG (X , e) a free Graev topological group?
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topological Nielsen-Schreier?
Nielsen-Schreier Theorem: Every subgroup of a free group is free.
Question: Is every subgroup H ≤ FG (X , e) a free Graev topological group?
No, not even if H is closed (Graev; Brown; Clarke; Hunt, Morris).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topological Nielsen-Schreier?
Nielsen-Schreier Theorem: Every subgroup of a free group is free.
Question: Is every subgroup H ≤ FG (X , e) a free Graev topological group?
No, not even if H is closed (Graev; Brown; Clarke; Hunt, Morris).
Yes, if X is Hausdorff kω -space and H is open (R. Brown, J.P. Hardy; P.
Nickolas).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topological Nielsen-Schreier?
Nielsen-Schreier Theorem: Every subgroup of a free group is free.
Question: Is every subgroup H ≤ FG (X , e) a free Graev topological group?
No, not even if H is closed (Graev; Brown; Clarke; Hunt, Morris).
Yes, if X is Hausdorff kω -space and H is open (R. Brown, J.P. Hardy; P.
Nickolas).
Yes, in abelian case AG (X , e), X Tychonoff and H open (Morris,
Pestov).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topological Nielsen-Schreier?
Nielsen-Schreier Theorem: Every subgroup of a free group is free.
Question: Is every subgroup H ≤ FG (X , e) a free Graev topological group?
No, not even if H is closed (Graev; Brown; Clarke; Hunt, Morris).
Yes, if X is Hausdorff kω -space and H is open (R. Brown, J.P. Hardy; P.
Nickolas).
Yes, in abelian case AG (X , e), X Tychonoff and H open (Morris,
Pestov).
Question: Is every open subgroup H ≤ FG (X , e) a free Graev topological
group? (at least for Tychonoff X )
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Main Results
Theorem 1: Every open subgroup of a free Graev topological group is
a free Graev topological group.
Theorem 2: Every open subgroup of a free Markov topological group is
a free Markov topological group iff it is disconnected.
*No separation axioms are required.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Main Results
Theorem 1: Every open subgroup of a free Graev topological group is
a free Graev topological group.
Theorem 2: Every open subgroup of a free Markov topological group is
a free Markov topological group iff it is disconnected.
*No separation axioms are required.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Putting algebraic topology to work
Algebraic topology: The fundamental group π1 and covering space theory
provide a straightforward proof of the Nielsen-Schreier Theorem.
Topology o
/ Algebra
Extension: Use a topologically enriched version πτ1 of the fundamental group
and a generalization of covering spaces (semicovering spaces).
“Wild” Topology o
Jeremy Brazas
/ Topological algebra
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Putting algebraic topology to work
Algebraic topology: The fundamental group π1 and covering space theory
provide a straightforward proof of the Nielsen-Schreier Theorem.
Topology o
/ Algebra
Extension: Use a topologically enriched version πτ1 of the fundamental group
and a generalization of covering spaces (semicovering spaces).
“Wild” Topology o
Jeremy Brazas
/ Topological algebra
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
The Nielsen-Schreier Theorem
Nielsen-Schreier Theorem: Every subgroup of a free group is free.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
The Nielsen-Schreier Theorem
Nielsen-Schreier Theorem: Every subgroup of a free group is free.
Lemma: The fundamental group of a graph is free.
'
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Completing the proof.
Proof.
S 1 , x0
W
There is a covering map p : Y → A S 1 such that
p∗ (π1 (Y , y0 )) = H.
Suppose H ≤ F(A )= π1
W
A
The covering of a graph is a graph, so Y is a graph.
Since Y is a graph, H π1 (Y , y0 ) is free.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Completing the proof.
Proof.
S 1 , x0
W
There is a covering map p : Y → A S 1 such that
p∗ (π1 (Y , y0 )) = H.
Suppose H ≤ F(A )= π1
W
A
The covering of a graph is a graph, so Y is a graph.
Since Y is a graph, H π1 (Y , y0 ) is free.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Completing the proof.
Proof.
S 1 , x0
W
There is a covering map p : Y → A S 1 such that
p∗ (π1 (Y , y0 )) = H.
Suppose H ≤ F(A )= π1
W
A
The covering of a graph is a graph, so Y is a graph.
Since Y is a graph, H π1 (Y , y0 ) is free.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Completing the proof.
Proof.
S 1 , x0
W
There is a covering map p : Y → A S 1 such that
p∗ (π1 (Y , y0 )) = H.
Suppose H ≤ F(A )= π1
W
A
The covering of a graph is a graph, so Y is a graph.
Since Y is a graph, H π1 (Y , y0 ) is free.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Completing the proof.
Proof.
S 1 , x0
W
There is a covering map p : Y → A S 1 such that
p∗ (π1 (Y , y0 )) = H.
Suppose H ≤ F(A )= π1
W
A
The covering of a graph is a graph, so Y is a graph.
Since Y is a graph, H π1 (Y , y0 ) is free.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Completing the proof.
Proof.
S 1 , x0
W
There is a covering map p : Y → A S 1 such that
p∗ (π1 (Y , y0 )) = H.
Suppose H ≤ F(A )= π1
W
A
The covering of a graph is a graph, so Y is a graph.
Since Y is a graph, H π1 (Y , y0 ) is free.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topologizing π1
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topologizing π1
Guiding principle: If αn → α in Ω(X , x), then [αn ] → [α] in π1 (X , x).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topologizing π1
Guiding principle: If αn → α in Ω(X , x), then [αn ] → [α] in π1 (X , x).
Ω(X , x) → π1 (X , x), α 7→ [α] should be continuous.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
The quasitopological fundamental group
Natural choice: Give π1 (X , x) the quotient topology with respect to
Ω(X , x) → π1 (X , x) (Biss).
I πqtop (X , x) is a quasitopological group (Calcut, McCarthy).
1
I πqtop (X , x) can fail to be a topological group even for Peano continua (P. Fabel).
1
I The topology of πqtop (X , x) is often complicated but can retain more local data
1
than shape invariants.
A left adjoint τ : qTopGrp → TopGrp removes the “fewest number” of open sets from
quasitopological G until a topological τ(G) is obtained.
Let πτ1 (X , x) = τ(π1
qtop
Jeremy Brazas
(X , x))
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
The quasitopological fundamental group
Natural choice: Give π1 (X , x) the quotient topology with respect to
Ω(X , x) → π1 (X , x) (Biss).
I πqtop (X , x) is a quasitopological group (Calcut, McCarthy).
1
I πqtop (X , x) can fail to be a topological group even for Peano continua (P. Fabel).
1
I The topology of πqtop (X , x) is often complicated but can retain more local data
1
than shape invariants.
A left adjoint τ : qTopGrp → TopGrp removes the “fewest number” of open sets from
quasitopological G until a topological τ(G) is obtained.
Let πτ1 (X , x) = τ(π1
qtop
Jeremy Brazas
(X , x))
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
The quasitopological fundamental group
Natural choice: Give π1 (X , x) the quotient topology with respect to
Ω(X , x) → π1 (X , x) (Biss).
I πqtop (X , x) is a quasitopological group (Calcut, McCarthy).
1
I πqtop (X , x) can fail to be a topological group even for Peano continua (P. Fabel).
1
I The topology of πqtop (X , x) is often complicated but can retain more local data
1
than shape invariants.
A left adjoint τ : qTopGrp → TopGrp removes the “fewest number” of open sets from
quasitopological G until a topological τ(G) is obtained.
Let πτ1 (X , x) = τ(π1
qtop
Jeremy Brazas
(X , x))
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
The quasitopological fundamental group
Natural choice: Give π1 (X , x) the quotient topology with respect to
Ω(X , x) → π1 (X , x) (Biss).
I πqtop (X , x) is a quasitopological group (Calcut, McCarthy).
1
I πqtop (X , x) can fail to be a topological group even for Peano continua (P. Fabel).
1
I The topology of πqtop (X , x) is often complicated but can retain more local data
1
than shape invariants.
A left adjoint τ : qTopGrp → TopGrp removes the “fewest number” of open sets from
quasitopological G until a topological τ(G) is obtained.
Let πτ1 (X , x) = τ(π1
qtop
Jeremy Brazas
(X , x))
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
The quasitopological fundamental group
Natural choice: Give π1 (X , x) the quotient topology with respect to
Ω(X , x) → π1 (X , x) (Biss).
I πqtop (X , x) is a quasitopological group (Calcut, McCarthy).
1
I πqtop (X , x) can fail to be a topological group even for Peano continua (P. Fabel).
1
I The topology of πqtop (X , x) is often complicated but can retain more local data
1
than shape invariants.
A left adjoint τ : qTopGrp → TopGrp removes the “fewest number” of open sets from
quasitopological G until a topological τ(G) is obtained.
Let πτ1 (X , x) = τ(π1
qtop
Jeremy Brazas
(X , x))
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topological π1
Characterizations:
I The topology of πτ (X , x) is the finest group topology on π1 (X , x) such
1
that Ω(X , x) → π1 (X , x) is continuous.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topological π1
Characterizations:
I The topology of πτ (X , x) is the finest group topology on π1 (X , x) such
1
that Ω(X , x) → π1 (X , x) is continuous.
I Transfinite approximation by quasitopological groups
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Topological π1
Characterizations:
I The topology of πτ (X , x) is the finest group topology on π1 (X , x) such
1
that Ω(X , x) → π1 (X , x) is continuous.
I Transfinite approximation by quasitopological groups
Utility: realizing universal topological groups
I πτ (Σ(Z+ )) FM (πqtop (Z)) where Σ(Z+ ) = Z × I/Z × {0, 1} is a
1
0
generalized wedge of circles.
I Topological van-Kampen theorems
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Semicovering maps
Definition: A map p : Y → X is a semicovering map if
I p is a local homeomorphism
I Whenever f is a path or homotopy of paths starting at p(y0 ) = x0 , there
is a unique lift e
f starting at y0
I Each lifting function f 7→ e
f is continuous on mapping spaces.
Every covering map is a semicovering map but not every semicovering is a
covering map.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Semicovering maps
Definition: A map p : Y → X is a semicovering map if
I p is a local homeomorphism
I Whenever f is a path or homotopy of paths starting at p(y0 ) = x0 , there
is a unique lift e
f starting at y0
I Each lifting function f 7→ e
f is continuous on mapping spaces.
Every covering map is a semicovering map but not every semicovering is a
covering map.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Semicovering maps
Definition: A map p : Y → X is a semicovering map if
I p is a local homeomorphism
I Whenever f is a path or homotopy of paths starting at p(y0 ) = x0 , there
is a unique lift e
f starting at y0
I Each lifting function f 7→ e
f is continuous on mapping spaces.
Every covering map is a semicovering map but not every semicovering is a
covering map.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Semicovering maps
Definition: A map p : Y → X is a semicovering map if
I p is a local homeomorphism
I Whenever f is a path or homotopy of paths starting at p(y0 ) = x0 , there
is a unique lift e
f starting at y0
I Each lifting function f 7→ e
f is continuous on mapping spaces.
Every covering map is a semicovering map but not every semicovering is a
covering map.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Semicovering maps
Definition: A map p : Y → X is a semicovering map if
I p is a local homeomorphism
I Whenever f is a path or homotopy of paths starting at p(y0 ) = x0 , there
is a unique lift e
f starting at y0
I Each lifting function f 7→ e
f is continuous on mapping spaces.
Every covering map is a semicovering map but not every semicovering is a
covering map.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Semicovering maps
Definition: A map p : Y → X is a semicovering map if
I p is a local homeomorphism
I Whenever f is a path or homotopy of paths starting at p(y0 ) = x0 , there
is a unique lift e
f starting at y0
I Each lifting function f 7→ e
f is continuous on mapping spaces.
Every covering map is a semicovering map but not every semicovering is a
covering map.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
A semicovering which is not a covering
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Traditional Covering Theory
A covering map p : Y → X induces an injection p∗ : π1 (Y , y0 ) → π1 (X , x0 ) of
groups. Let H = p∗ (π1 (Y , y0 )).
Classification of covering maps: If X is locally “nice” (locally path
connected, and semilocally simply connected), then there is a bijective
correspondence
(
)
Equivalence classes of o
coverings p : Y → X
Jeremy Brazas
/
(
Conjugacy classes of
subgroups H ≤ π1 (X , x)
)
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Semicovering Space Theory
A semicovering map p : Y → X induces an open embedding
p∗ : πτ1 (Y , y0 ) → πτ1 (X , x0 ) of topological groups. Let H = p∗ (πτ1 (Y , y0 )).
Classification of semicovering maps: If X is locally wep-connected, then
there is a bijective correspondence
(
)
Equivalence classes of
semicoverings p : Y → X
o
Jeremy Brazas
/
(
)
Conjugacy classes of
open subgroups H ≤ πτ1 (X , x)
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Top-graphs
Replacement of graphs: A Top-graph Γ consists of a discrete space of vertices and an
edge space Γ(x, y) for each ordered pair of vertices (x, y).
Lemma: If Γ is a Top-graph, then πτ1 (Γ, x) is a free Graev topological group.
Proof. Use universal constructions of topologically enriched categories and groupoids
(i.e. Top-categories and Top-groupoids).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Top-graphs
Replacement of graphs: A Top-graph Γ consists of a discrete space of vertices and an
edge space Γ(x, y) for each ordered pair of vertices (x, y).
Lemma: If Γ is a Top-graph, then πτ1 (Γ, x) is a free Graev topological group.
Proof. Use universal constructions of topologically enriched categories and groupoids
(i.e. Top-categories and Top-groupoids).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Top-graphs
Replacement of graphs: A Top-graph Γ consists of a discrete space of vertices and an
edge space Γ(x, y) for each ordered pair of vertices (x, y).
Lemma: If Γ is a Top-graph, then πτ1 (Γ, x) is a free Graev topological group.
Proof. Use universal constructions of topologically enriched categories and groupoids
(i.e. Top-categories and Top-groupoids).
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Proof sketch
Suppose H ≤ FG (X , e) is open.
Find Z such that πqtop
(Z) X (D. Harris)
0
The unreduced suspension Γ = SZ is a Top-graph such that
πτ1 (Γ, x) = FG (X , e).
There is a semicovering p : Y → Γ such that p∗ (πτ1 (Y , y)) = H.
A semicovering of a Top-graph is a Top-graph, so Y is a Top-graph.
Since Y is a Top-graph, p∗ : πτ1 (Y , y) H is free Graev topological.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Proof sketch
Suppose H ≤ FG (X , e) is open.
Find Z such that πqtop
(Z) X (D. Harris)
0
The unreduced suspension Γ = SZ is a Top-graph such that
πτ1 (Γ, x) = FG (X , e).
There is a semicovering p : Y → Γ such that p∗ (πτ1 (Y , y)) = H.
A semicovering of a Top-graph is a Top-graph, so Y is a Top-graph.
Since Y is a Top-graph, p∗ : πτ1 (Y , y) H is free Graev topological.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Proof sketch
Suppose H ≤ FG (X , e) is open.
Find Z such that πqtop
(Z) X (D. Harris)
0
The unreduced suspension Γ = SZ is a Top-graph such that
πτ1 (Γ, x) = FG (X , e).
There is a semicovering p : Y → Γ such that p∗ (πτ1 (Y , y)) = H.
A semicovering of a Top-graph is a Top-graph, so Y is a Top-graph.
Since Y is a Top-graph, p∗ : πτ1 (Y , y) H is free Graev topological.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Proof sketch
Suppose H ≤ FG (X , e) is open.
Find Z such that πqtop
(Z) X (D. Harris)
0
The unreduced suspension Γ = SZ is a Top-graph such that
πτ1 (Γ, x) = FG (X , e).
There is a semicovering p : Y → Γ such that p∗ (πτ1 (Y , y)) = H.
A semicovering of a Top-graph is a Top-graph, so Y is a Top-graph.
Since Y is a Top-graph, p∗ : πτ1 (Y , y) H is free Graev topological.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Proof sketch
Suppose H ≤ FG (X , e) is open.
Find Z such that πqtop
(Z) X (D. Harris)
0
The unreduced suspension Γ = SZ is a Top-graph such that
πτ1 (Γ, x) = FG (X , e).
There is a semicovering p : Y → Γ such that p∗ (πτ1 (Y , y)) = H.
A semicovering of a Top-graph is a Top-graph, so Y is a Top-graph.
Since Y is a Top-graph, p∗ : πτ1 (Y , y) H is free Graev topological.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
Proof sketch
Suppose H ≤ FG (X , e) is open.
Find Z such that πqtop
(Z) X (D. Harris)
0
The unreduced suspension Γ = SZ is a Top-graph such that
πτ1 (Γ, x) = FG (X , e).
There is a semicovering p : Y → Γ such that p∗ (πτ1 (Y , y)) = H.
A semicovering of a Top-graph is a Top-graph, so Y is a Top-graph.
Since Y is a Top-graph, p∗ : πτ1 (Y , y) H is free Graev topological.
Jeremy Brazas
Open subgroups of free topological groups
Free topological groups
Topological π1
Semicoverings
Proof Sketch
References
J. Brazas, Semicoverings: A generalization of covering space theory. Homology Homotopy Appl. 14 (2012)
33-63.
J. Brazas, The fundamental group as topological group. Topology Appl. 160 (2013) 170-188.
J. Brazas, Open subgroups of free topological groups. Submitted. 2013. arXiv:1209.5486.
R. Brown, J.P.L. Hardy, Subgroups of free topological groups and free topological products of topological
groups, J. London Math. Soc. (2) 10 (1975) 431-440.
P. Fabel, Multiplication is discontinuous in the Hawaiian earring group. Bull. Polish Acad. Sci. Math. 59 (2011)
7783.
Graev, M.I. Free topological groups. Amer. Math. Soc. Transl. 8 (1962) 305-365.
Markov, A.A. On free topological groups. Izv. Akad. Nauk. SSSR Ser. Mat. 9 (1945) 3-64.
S.A. Morris, V.G. Pestov, Open subgroups of free abelian topological groups. Math. Proc. Camb. Phil. Soc.
114 (1993) 439-442.
P. Nickolas, A Schreier theorem for free topological groups, Bulletin of the Australian Math. Soc. 13 (1975)
121–127.
O. Schreier, Die Untergruppen der freien Gruppen, Abh. Mat Sem. Univ. Hamburg 3 (1927) 167-169.
Jeremy Brazas
Open subgroups of free topological groups