Homotopy Groups of Continua as Topological Group Shapes, quotients, - - PowerPoint PPT Presentation

Homotopy Groups of Continua as Topological Group

Shapes, quotients, and a clash of two categories Paul Fabel

Mississippi State University

July 2013

P.F. (Institute) Beamer presentations in SWP and SW July 2013 1 / 14

Why should we care?

Why could it be useful to consider πn(X, p) as topological space or topological group?

P.F. (Institute) Beamer presentations in SWP and SW July 2013 2 / 14

Why should we care?

Why could it be useful to consider πn(X, p) as topological space or topological group? If X is locally complicated πn(X, p) often ‘wants’ to have an interesting topology so that the topology of πn(X, p) is an invariant

• f X itself.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 2 / 14

Why should we care?

Why could it be useful to consider πn(X, p) as topological space or topological group? If X is locally complicated πn(X, p) often ‘wants’ to have an interesting topology so that the topology of πn(X, p) is an invariant

• f X itself.

In particular if πn(X, p) is isomorphic to πn(Y , q) we can hope to distinguish X and Y by asking if πn(X, p) is homeomorphic or not to πn(Y , q).

P.F. (Institute) Beamer presentations in SWP and SW July 2013 2 / 14

Topology on homotopy groups of a continuum

Given a continuum X, what are some strategies for imposing topology

• n the homotopy groups πn(X, p)?

P.F. (Institute) Beamer presentations in SWP and SW July 2013 3 / 14

Topology on homotopy groups of a continuum

Given a continuum X, what are some strategies for imposing topology

• n the homotopy groups πn(X, p)?

Try to use topological quotients in a natural manner.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 3 / 14

Topology on homotopy groups of a continuum

Given a continuum X, what are some strategies for imposing topology

• n the homotopy groups πn(X, p)?

Try to use topological quotients in a natural manner. Try to use metric quotients or pseudo metric quotients in a natural manner.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 3 / 14

Topology on homotopy groups of a continuum

Given a continuum X, what are some strategies for imposing topology

• n the homotopy groups πn(X, p)?

Try to use topological quotients in a natural manner. Try to use metric quotients or pseudo metric quotients in a natural manner. Try to use shape theory in a natural manner.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 3 / 14

Topology on homotopy groups of a continuum

Given a continuum X, what are some strategies for imposing topology

• n the homotopy groups πn(X, p)?

Try to use topological quotients in a natural manner. Try to use metric quotients or pseudo metric quotients in a natural manner. Try to use shape theory in a natural manner. We will make these answers more precise soon

P.F. (Institute) Beamer presentations in SWP and SW July 2013 3 / 14

What we will achieve

What we will attempt to convey in this talk:

P.F. (Institute) Beamer presentations in SWP and SW July 2013 4 / 14

What we will achieve

What we will attempt to convey in this talk: We discuss 3 distinct topologies on πn(X, p), each of which is an invariant of homotopy type the continuum.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 4 / 14

What we will achieve

What we will attempt to convey in this talk: We discuss 3 distinct topologies on πn(X, p), each of which is an invariant of homotopy type the continuum. The pseudo metric quotients will have strong ties to shape theory, but in a natural sense proves to be a sharper tool

P.F. (Institute) Beamer presentations in SWP and SW July 2013 4 / 14

What we will achieve

What we will attempt to convey in this talk: We discuss 3 distinct topologies on πn(X, p), each of which is an invariant of homotopy type the continuum. The pseudo metric quotients will have strong ties to shape theory, but in a natural sense proves to be a sharper tool The quotient topology proves sharper still but often at the cost of metrizability.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 4 / 14

What we will achieve

What we will attempt to convey in this talk: We discuss 3 distinct topologies on πn(X, p), each of which is an invariant of homotopy type the continuum. The pseudo metric quotients will have strong ties to shape theory, but in a natural sense proves to be a sharper tool The quotient topology proves sharper still but often at the cost of metrizability. However the quotient topology often has the capacity to distinguish homotopy type when the other methods fail.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 4 / 14

What we will achieve

What we will attempt to convey in this talk: We discuss 3 distinct topologies on πn(X, p), each of which is an invariant of homotopy type the continuum. The pseudo metric quotients will have strong ties to shape theory, but in a natural sense proves to be a sharper tool The quotient topology proves sharper still but often at the cost of metrizability. However the quotient topology often has the capacity to distinguish homotopy type when the other methods fail. Planar and other low dimensional Peano continua illustrate the meaning and usefulness of the 3 de…ntions/tools.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 4 / 14

What we will achieve

What we will attempt to convey in this talk: We discuss 3 distinct topologies on πn(X, p), each of which is an invariant of homotopy type the continuum. The pseudo metric quotients will have strong ties to shape theory, but in a natural sense proves to be a sharper tool The quotient topology proves sharper still but often at the cost of metrizability. However the quotient topology often has the capacity to distinguish homotopy type when the other methods fail. Planar and other low dimensional Peano continua illustrate the meaning and usefulness of the 3 de…ntions/tools. πn(X, p) with quotient topology accentuates a fundamental shortcoming in the general de…nition of product topology of G H, making the case for example, for the relevance and utility of the category of sequential spaces SEQ.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 4 / 14

Familiar or easy de…nitions:

What is a pseudo metric?

P.F. (Institute) Beamer presentations in SWP and SW July 2013 5 / 14

Familiar or easy de…nitions:

What is a pseudo metric? (A metric except D(x, y) = 0 is permitted if x 6= y)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 5 / 14

Familiar or easy de…nitions:

What is a pseudo metric? (A metric except D(x, y) = 0 is permitted if x 6= y) A function D : Y Y ! [0, ∞) such that

P.F. (Institute) Beamer presentations in SWP and SW July 2013 5 / 14

Familiar or easy de…nitions:

What is a pseudo metric? (A metric except D(x, y) = 0 is permitted if x 6= y) A function D : Y Y ! [0, ∞) such that D(x, y) = D(y, x)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 5 / 14

Familiar or easy de…nitions:

What is a pseudo metric? (A metric except D(x, y) = 0 is permitted if x 6= y) A function D : Y Y ! [0, ∞) such that D(x, y) = D(y, x) D(x, x) = 0

P.F. (Institute) Beamer presentations in SWP and SW July 2013 5 / 14

Familiar or easy de…nitions:

What is a pseudo metric? (A metric except D(x, y) = 0 is permitted if x 6= y) A function D : Y Y ! [0, ∞) such that D(x, y) = D(y, x) D(x, x) = 0 D(x, y) + D(y, z) D(x, z)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 5 / 14

Familiar or easy de…nitions:

What is a pseudo metric? (A metric except D(x, y) = 0 is permitted if x 6= y) A function D : Y Y ! [0, ∞) such that D(x, y) = D(y, x) D(x, x) = 0 D(x, y) + D(y, z) D(x, z) Every pseudometric space generates a canonical metric (Kolmogorov) quotient, x~y i¤ D(x, y) = 0

P.F. (Institute) Beamer presentations in SWP and SW July 2013 5 / 14

Two natural quotients spaces

Two natural quotients

P.F. (Institute) Beamer presentations in SWP and SW July 2013 6 / 14

Two natural quotients spaces

Two natural quotients Every equivalence relation on a metric space (Y , d) generates two generally distinct topologies on the equivalence classes [y] 2 Y . Let q : Y ! Y denote the natural function q(y) = [y].

P.F. (Institute) Beamer presentations in SWP and SW July 2013 6 / 14

Two natural quotients spaces

Two natural quotients Every equivalence relation on a metric space (Y , d) generates two generally distinct topologies on the equivalence classes [y] 2 Y . Let q : Y ! Y denote the natural function q(y) = [y]. The quotient topology: A Y is closed i¤ q1(A) Y closed

P.F. (Institute) Beamer presentations in SWP and SW July 2013 6 / 14

Two natural quotients spaces

Two natural quotients Every equivalence relation on a metric space (Y , d) generates two generally distinct topologies on the equivalence classes [y] 2 Y . Let q : Y ! Y denote the natural function q(y) = [y]. The quotient topology: A Y is closed i¤ q1(A) Y closed The pseudo metric quotient (Y , D) generated by the condition D([x], [y]) < ε if d(x^, y^) < ε for some x^ 2 [x] and y^ 2 [y].

P.F. (Institute) Beamer presentations in SWP and SW July 2013 6 / 14

Two natural quotients spaces

Two natural quotients Every equivalence relation on a metric space (Y , d) generates two generally distinct topologies on the equivalence classes [y] 2 Y . Let q : Y ! Y denote the natural function q(y) = [y]. The quotient topology: A Y is closed i¤ q1(A) Y closed The pseudo metric quotient (Y , D) generated by the condition D([x], [y]) < ε if d(x^, y^) < ε for some x^ 2 [x] and y^ 2 [y]. Precisely D([x], [y]) < ε i¤ there exists a …nite sequence [x] = [x0], [x1], ...[xK ] = [y] and yi 2 [xi] so that ΣN1

i=0 d(yi, xi+1) < ε

P.F. (Institute) Beamer presentations in SWP and SW July 2013 6 / 14

Two natural quotients spaces

Two natural quotients Every equivalence relation on a metric space (Y , d) generates two generally distinct topologies on the equivalence classes [y] 2 Y . Let q : Y ! Y denote the natural function q(y) = [y]. The quotient topology: A Y is closed i¤ q1(A) Y closed The pseudo metric quotient (Y , D) generated by the condition D([x], [y]) < ε if d(x^, y^) < ε for some x^ 2 [x] and y^ 2 [y]. Precisely D([x], [y]) < ε i¤ there exists a …nite sequence [x] = [x0], [x1], ...[xK ] = [y] and yi 2 [xi] so that ΣN1

i=0 d(yi, xi+1) < ε

If Y is compact and Y is T2, then the quotients coincide.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 6 / 14

Two natural quotients spaces

Two natural quotients Every equivalence relation on a metric space (Y , d) generates two generally distinct topologies on the equivalence classes [y] 2 Y . Let q : Y ! Y denote the natural function q(y) = [y]. The quotient topology: A Y is closed i¤ q1(A) Y closed The pseudo metric quotient (Y , D) generated by the condition D([x], [y]) < ε if d(x^, y^) < ε for some x^ 2 [x] and y^ 2 [y]. Precisely D([x], [y]) < ε i¤ there exists a …nite sequence [x] = [x0], [x1], ...[xK ] = [y] and yi 2 [xi] so that ΣN1

i=0 d(yi, xi+1) < ε

If Y is compact and Y is T2, then the quotients coincide. Taking Y = f0, ...1

4, 1 3, 1 2, 1g and identifying 1 m ~ 1 n shows why we need

T2.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 6 / 14

Two natural quotients spaces

Two natural quotients Every equivalence relation on a metric space (Y , d) generates two generally distinct topologies on the equivalence classes [y] 2 Y . Let q : Y ! Y denote the natural function q(y) = [y]. The quotient topology: A Y is closed i¤ q1(A) Y closed The pseudo metric quotient (Y , D) generated by the condition D([x], [y]) < ε if d(x^, y^) < ε for some x^ 2 [x] and y^ 2 [y]. Precisely D([x], [y]) < ε i¤ there exists a …nite sequence [x] = [x0], [x1], ...[xK ] = [y] and yi 2 [xi] so that ΣN1

i=0 d(yi, xi+1) < ε

If Y is compact and Y is T2, then the quotients coincide. Taking Y = f0, ...1

4, 1 3, 1 2, 1g and identifying 1 m ~ 1 n shows why we need

T2. Glue together countably many copies of [0, 1] at 0, yields distinct T2 quotients.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 6 / 14

Three topologies

How exactly can we impose topology on πn(X, p) if X is a continuum?

P.F. (Institute) Beamer presentations in SWP and SW July 2013 7 / 14

Three topologies

How exactly can we impose topology on πn(X, p) if X is a continuum? There are (at least) 3 natural ways.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 7 / 14

Three topologies

How exactly can we impose topology on πn(X, p) if X is a continuum? There are (at least) 3 natural ways. Start with the space C(Sn, 1), (X, p) of based maps of the sphere Sn ! X

P.F. (Institute) Beamer presentations in SWP and SW July 2013 7 / 14

Three topologies

How exactly can we impose topology on πn(X, p) if X is a continuum? There are (at least) 3 natural ways. Start with the space C(Sn, 1), (X, p) of based maps of the sphere Sn ! X Impose the uniform metric on C(Sn, 1), (X, p).

P.F. (Institute) Beamer presentations in SWP and SW July 2013 7 / 14

Three topologies

How exactly can we impose topology on πn(X, p) if X is a continuum? There are (at least) 3 natural ways. Start with the space C(Sn, 1), (X, p) of based maps of the sphere Sn ! X Impose the uniform metric on C(Sn, 1), (X, p). Collapse each path component of C(Sn, 1), (X, p) to a point, to create the set πn(X, p).

P.F. (Institute) Beamer presentations in SWP and SW July 2013 7 / 14

Three topologies

How exactly can we impose topology on πn(X, p) if X is a continuum? There are (at least) 3 natural ways. Start with the space C(Sn, 1), (X, p) of based maps of the sphere Sn ! X Impose the uniform metric on C(Sn, 1), (X, p). Collapse each path component of C(Sn, 1), (X, p) to a point, to create the set πn(X, p). We have natural surjection q : C(Sn, 1), (X, p) ! πn(X, p)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 7 / 14

Three topologies

How exactly can we impose topology on πn(X, p) if X is a continuum? There are (at least) 3 natural ways. Start with the space C(Sn, 1), (X, p) of based maps of the sphere Sn ! X Impose the uniform metric on C(Sn, 1), (X, p). Collapse each path component of C(Sn, 1), (X, p) to a point, to create the set πn(X, p). We have natural surjection q : C(Sn, 1), (X, p) ! πn(X, p) Impose the quotient topology on πquotient

n

(X, p)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 7 / 14

Three topologies

How exactly can we impose topology on πn(X, p) if X is a continuum? There are (at least) 3 natural ways. Start with the space C(Sn, 1), (X, p) of based maps of the sphere Sn ! X Impose the uniform metric on C(Sn, 1), (X, p). Collapse each path component of C(Sn, 1), (X, p) to a point, to create the set πn(X, p). We have natural surjection q : C(Sn, 1), (X, p) ! πn(X, p) Impose the quotient topology on πquotient

n

(X, p) OR

P.F. (Institute) Beamer presentations in SWP and SW July 2013 7 / 14

Three topologies

How exactly can we impose topology on πn(X, p) if X is a continuum? There are (at least) 3 natural ways. Start with the space C(Sn, 1), (X, p) of based maps of the sphere Sn ! X Impose the uniform metric on C(Sn, 1), (X, p). Collapse each path component of C(Sn, 1), (X, p) to a point, to create the set πn(X, p). We have natural surjection q : C(Sn, 1), (X, p) ! πn(X, p) Impose the quotient topology on πquotient

n

(X, p) OR impose the pseudo-metric quotient on πpseudometric

n

(X, p).

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We can also employ shape theory

We can also employ shape theory to create πshape

n

(X, p)

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We can also employ shape theory

We can also employ shape theory to create πshape

n

(X, p) Pull back the shape group homomorphism φ : πn(X, p) ! lim πn(Um, p)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 8 / 14

We can also employ shape theory

We can also employ shape theory to create πshape

n

(X, p) Pull back the shape group homomorphism φ : πn(X, p) ! lim πn(Um, p) Declare φ1(V ) πn(X, p) open i¤ V lim πn(Um, p) is open.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 8 / 14

We can also employ shape theory

We can also employ shape theory to create πshape

n

(X, p) Pull back the shape group homomorphism φ : πn(X, p) ! lim πn(Um, p) Declare φ1(V ) πn(X, p) open i¤ V lim πn(Um, p) is open. (If don’t know much shape theory, embed X l2, let Um be the union of …nitely many

1 2m open balls covering X, arrange Un+1 Un,

φ is induced by j : X ! lim Un with inclusion bonding maps).

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How do the 3 topologies compare?

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How do the 3 topologies compare? If X is a continuum we get a nice answer

P.F. (Institute) Beamer presentations in SWP and SW July 2013 9 / 14

How do the 3 topologies compare? If X is a continuum we get a nice answer They re…ne each other.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 9 / 14

How do the 3 topologies compare? If X is a continuum we get a nice answer They re…ne each other. i.e. id is continuous as follows.

P.F. (Institute) Beamer presentations in SWP and SW July 2013 9 / 14

How do the 3 topologies compare? If X is a continuum we get a nice answer They re…ne each other. i.e. id is continuous as follows. id : πquotient

n

(X, p) ! πpseudometric

n

(X, p) ! πshape

n

(X, p)

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The Peano continuum X =Hawaiian earring shows

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The Peano continuum X =Hawaiian earring shows The continuous isomorphism id : πquotient

n

(X, p) ! πpseudometric

n

(X, p)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 10 / 14

The Peano continuum X =Hawaiian earring shows The continuous isomorphism id : πquotient

n

(X, p) ! πpseudometric

n

(X, p) might NOT be a homeomorphism ([F] 2005 AGT)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 10 / 14

The Peano continuum X =Hawaiian earring shows The continuous isomorphism id : πquotient

n

(X, p) ! πpseudometric

n

(X, p) might NOT be a homeomorphism ([F] 2005 AGT) In fact π1(HE, p) is not a topological group in TOP.

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What are the theorems?

If X is the inverse limit of nested compact polyhedral retracts then...

P.F. (Institute) Beamer presentations in SWP and SW July 2013 11 / 14

What are the theorems?

If X is the inverse limit of nested compact polyhedral retracts then... πpseudometric

n

(X, p)~πshape

n

(X, p)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 11 / 14

What are the theorems?

If X is the inverse limit of nested compact polyhedral retracts then... πpseudometric

n

(X, p)~πshape

n

(X, p) In particular if X is the inverse limit of nested compact polyhedral retracts TFAE

P.F. (Institute) Beamer presentations in SWP and SW July 2013 11 / 14

What are the theorems?

If X is the inverse limit of nested compact polyhedral retracts then... πpseudometric

n

(X, p)~πshape

n

(X, p) In particular if X is the inverse limit of nested compact polyhedral retracts TFAE X is πn shape injective

P.F. (Institute) Beamer presentations in SWP and SW July 2013 11 / 14

What are the theorems?

If X is the inverse limit of nested compact polyhedral retracts then... πpseudometric

n

(X, p)~πshape

n

(X, p) In particular if X is the inverse limit of nested compact polyhedral retracts TFAE X is πn shape injective πpseudometric

n

(X, p) is a metric space

P.F. (Institute) Beamer presentations in SWP and SW July 2013 11 / 14

A 2-dimensional Peano continuum shows they are distinct?

A 2-dimensional Peano continuum shows the 3 topologies are distinct?

P.F. (Institute) Beamer presentations in SWP and SW July 2013 12 / 14

A 2-dimensional Peano continuum shows they are distinct?

A 2-dimensional Peano continuum shows the 3 topologies are distinct? No!

P.F. (Institute) Beamer presentations in SWP and SW July 2013 12 / 14

A 2-dimensional Peano continuum shows they are distinct?

A 2-dimensional Peano continuum shows the 3 topologies are distinct? No! Apparent Theorem ( Yesterday afternoon stroll) [Brazas] [F])

P.F. (Institute) Beamer presentations in SWP and SW July 2013 12 / 14

A 2-dimensional Peano continuum shows they are distinct?

A 2-dimensional Peano continuum shows the 3 topologies are distinct? No! Apparent Theorem ( Yesterday afternoon stroll) [Brazas] [F]) Suppose X is a metric Peano continuum, then

P.F. (Institute) Beamer presentations in SWP and SW July 2013 12 / 14

A 2-dimensional Peano continuum shows they are distinct?

A 2-dimensional Peano continuum shows the 3 topologies are distinct? No! Apparent Theorem ( Yesterday afternoon stroll) [Brazas] [F]) Suppose X is a metric Peano continuum, then πpseudometric

1

(X, p)~πshape

1

(X, p)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 12 / 14

A 2-dimensional Peano continuum shows they are distinct?

A 2-dimensional Peano continuum shows the 3 topologies are distinct? No! Apparent Theorem ( Yesterday afternoon stroll) [Brazas] [F]) Suppose X is a metric Peano continuum, then πpseudometric

1

(X, p)~πshape

1

(X, p) Follows from main results in

P.F. (Institute) Beamer presentations in SWP and SW July 2013 12 / 14

A 2-dimensional Peano continuum shows they are distinct?

A 2-dimensional Peano continuum shows the 3 topologies are distinct? No! Apparent Theorem ( Yesterday afternoon stroll) [Brazas] [F]) Suppose X is a metric Peano continuum, then πpseudometric

1

(X, p)~πshape

1

(X, p) Follows from main results in Thick Spanier groups and the …rst shape group ([Brazas][F])

P.F. (Institute) Beamer presentations in SWP and SW July 2013 12 / 14

A 2-dimensional Peano continuum shows they are distinct?

A 2-dimensional Peano continuum shows the 3 topologies are distinct? No! Apparent Theorem ( Yesterday afternoon stroll) [Brazas] [F]) Suppose X is a metric Peano continuum, then πpseudometric

1

(X, p)~πshape

1

(X, p) Follows from main results in Thick Spanier groups and the …rst shape group ([Brazas][F]) (To appear Rocky Mountain Journal of Mathematics)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 12 / 14

A 2-dimensional Peano continuum shows they are distinct?

A 2-dimensional Peano continuum shows the 3 topologies are distinct? No! Apparent Theorem ( Yesterday afternoon stroll) [Brazas] [F]) Suppose X is a metric Peano continuum, then πpseudometric

1

(X, p)~πshape

1

(X, p) Follows from main results in Thick Spanier groups and the …rst shape group ([Brazas][F]) (To appear Rocky Mountain Journal of Mathematics) Moral: If X is a Peano continuum the image of π1(X, p) in the …rst shape group can be understood intrinsically and geometrically without reference to open covers of X

P.F. (Institute) Beamer presentations in SWP and SW July 2013 12 / 14

What are the theorems?

If X is the inverse limit of nested compact polyhedral retracts then...

P.F. (Institute) Beamer presentations in SWP and SW July 2013 13 / 14

What are the theorems?

If X is the inverse limit of nested compact polyhedral retracts then... πpseudometric

n

(X, p)~πshape

n

(X, p)

P.F. (Institute) Beamer presentations in SWP and SW July 2013 13 / 14

What are the theorems?

If X is the inverse limit of nested compact polyhedral retracts then... πpseudometric

n

(X, p)~πshape

n

(X, p) In particular if X is the inverse limit of nested compact polyhedral retracts TFAE

P.F. (Institute) Beamer presentations in SWP and SW July 2013 13 / 14

What are the theorems?

If X is the inverse limit of nested compact polyhedral retracts then... πpseudometric

n

(X, p)~πshape

n

(X, p) In particular if X is the inverse limit of nested compact polyhedral retracts TFAE X is πn shape injective

P.F. (Institute) Beamer presentations in SWP and SW July 2013 13 / 14

What are the theorems?

If X is the inverse limit of nested compact polyhedral retracts then... πpseudometric

n

(X, p)~πshape

n

(X, p) In particular if X is the inverse limit of nested compact polyhedral retracts TFAE X is πn shape injective πpseudometric

n

(X, p) is a metric space

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What are the interesting examples?

What are the interesting examples?

P.F. (Institute) Beamer presentations in SWP and SW July 2013 14 / 14

What are the interesting examples?

What are the interesting examples? The punctured plane X = R2nf(0, 0)g.

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What are the interesting examples?

What are the interesting examples? The punctured plane X = R2nf(0, 0)g. It is locally compact but the topology of π1(X, p) depends on the metric of X

P.F. (Institute) Beamer presentations in SWP and SW July 2013 14 / 14

What are the interesting examples?

What are the interesting examples? The punctured plane X = R2nf(0, 0)g. It is locally compact but the topology of π1(X, p) depends on the metric of X This is why, to get a nice theory, it is helpful to assume X is a compact metric space or continuum

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