Download CW-complexes in the category of exterior spaces

Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
CW-complexes in the category of exterior spaces
J.M. García Calcines1
1
2
P.R. García Díaz1
A. Murillo Mas2
Departamento de Matemática Fundamental
Universidad de La Laguna
Departamento de Álgebra, Geometría y Topología
Universidad de Málaga
June 22-28 /International Category Theory Conference 2008. Calais,
France.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Introduction.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Introduction.
Ordinary homotopy invariants do not faithfully reflect the behavior and
geometry of non compact spaces at infinity.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Introduction.
Ordinary homotopy invariants do not faithfully reflect the behavior and
geometry of non compact spaces at infinity.
Proper homotopy theory
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Introduction.
Ordinary homotopy invariants do not faithfully reflect the behavior and
geometry of non compact spaces at infinity.
Proper homotopy theory
However, in the proper category we cannot develop many homotopy
constructions, such as loop spaces or homotopy fibers, since
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Introduction.
Ordinary homotopy invariants do not faithfully reflect the behavior and
geometry of non compact spaces at infinity.
Proper homotopy theory
However, in the proper category we cannot develop many homotopy
constructions, such as loop spaces or homotopy fibers, since
1
There are few limits and colimits.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Introduction.
Ordinary homotopy invariants do not faithfully reflect the behavior and
geometry of non compact spaces at infinity.
Proper homotopy theory
However, in the proper category we cannot develop many homotopy
constructions, such as loop spaces or homotopy fibers, since
1
2
There are few limits and colimits.
There is not a notion of fibration.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Introduction.
Ordinary homotopy invariants do not faithfully reflect the behavior and
geometry of non compact spaces at infinity.
Proper homotopy theory
However, in the proper category we cannot develop many homotopy
constructions, such as loop spaces or homotopy fibers, since
1
2
There are few limits and colimits.
There is not a notion of fibration.
A useful technique which avoid these problems is to embed the proper
category into a complete and cocomplete category and to use homotopy
theories that assume the existence of limits and colimits.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Introduction.
Ordinary homotopy invariants do not faithfully reflect the behavior and
geometry of non compact spaces at infinity.
Proper homotopy theory
However, in the proper category we cannot develop many homotopy
constructions, such as loop spaces or homotopy fibers, since
1
2
There are few limits and colimits.
There is not a notion of fibration.
A useful technique which avoid these problems is to embed the proper
category into a complete and cocomplete category and to use homotopy
theories that assume the existence of limits and colimits.
We have the Edwards-Hastings embedding of the proper homotopy
category of locally compact, σ-compact Hausdorff spaces into the
homotopy category of pro-spaces.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Introduction.
Ordinary homotopy invariants do not faithfully reflect the behavior and
geometry of non compact spaces at infinity.
Proper homotopy theory
However, in the proper category we cannot develop many homotopy
constructions, such as loop spaces or homotopy fibers, since
1
2
There are few limits and colimits.
There is not a notion of fibration.
A useful technique which avoid these problems is to embed the proper
category into a complete and cocomplete category and to use homotopy
theories that assume the existence of limits and colimits.
We have the Edwards-Hastings embedding of the proper homotopy
category of locally compact, σ-compact Hausdorff spaces into the
homotopy category of pro-spaces.
1
One has a strong restriction
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Introduction.
Ordinary homotopy invariants do not faithfully reflect the behavior and
geometry of non compact spaces at infinity.
Proper homotopy theory
However, in the proper category we cannot develop many homotopy
constructions, such as loop spaces or homotopy fibers, since
1
2
There are few limits and colimits.
There is not a notion of fibration.
A useful technique which avoid these problems is to embed the proper
category into a complete and cocomplete category and to use homotopy
theories that assume the existence of limits and colimits.
We have the Edwards-Hastings embedding of the proper homotopy
category of locally compact, σ-compact Hausdorff spaces into the
homotopy category of pro-spaces.
1
2
One has a strong restriction
The homotopy constructions produce pro-spaces that many times can not
be geometrically interpreted as regular spaces.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
An alternative category is that of the exterior spaces, introduced in
1998:
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
An alternative category is that of the exterior spaces, introduced in
1998:
J.M. Garcia-Calcines, M. Garcia-Pinillos, L.J. Hernandez Paricio. A
closed simplicial model category for proper homotopy and shape
theories. Bull. Austral. Math. Soc., vol. 57 (1998), 221-242.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
An alternative category is that of the exterior spaces, introduced in
1998:
J.M. Garcia-Calcines, M. Garcia-Pinillos, L.J. Hernandez Paricio. A
closed simplicial model category for proper homotopy and shape
theories. Bull. Austral. Math. Soc., vol. 57 (1998), 221-242.
(Roughly speaking, an exterior space is a topological space with a
“neighborhood system at infinity")
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
An alternative category is that of the exterior spaces, introduced in
1998:
J.M. Garcia-Calcines, M. Garcia-Pinillos, L.J. Hernandez Paricio. A
closed simplicial model category for proper homotopy and shape
theories. Bull. Austral. Math. Soc., vol. 57 (1998), 221-242.
(Roughly speaking, an exterior space is a topological space with a
“neighborhood system at infinity")
The notion of exterior space is established in such a way that
P⊂E
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
An alternative category is that of the exterior spaces, introduced in
1998:
J.M. Garcia-Calcines, M. Garcia-Pinillos, L.J. Hernandez Paricio. A
closed simplicial model category for proper homotopy and shape
theories. Bull. Austral. Math. Soc., vol. 57 (1998), 221-242.
(Roughly speaking, an exterior space is a topological space with a
“neighborhood system at infinity")
The notion of exterior space is established in such a way that
P⊂E
1
E is complete and cocomplete
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
An alternative category is that of the exterior spaces, introduced in
1998:
J.M. Garcia-Calcines, M. Garcia-Pinillos, L.J. Hernandez Paricio. A
closed simplicial model category for proper homotopy and shape
theories. Bull. Austral. Math. Soc., vol. 57 (1998), 221-242.
(Roughly speaking, an exterior space is a topological space with a
“neighborhood system at infinity")
The notion of exterior space is established in such a way that
P⊂E
1
2
E is complete and cocomplete
E has a model category structure in the sense of Quillen.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
An alternative category is that of the exterior spaces, introduced in
1998:
J.M. Garcia-Calcines, M. Garcia-Pinillos, L.J. Hernandez Paricio. A
closed simplicial model category for proper homotopy and shape
theories. Bull. Austral. Math. Soc., vol. 57 (1998), 221-242.
(Roughly speaking, an exterior space is a topological space with a
“neighborhood system at infinity")
The notion of exterior space is established in such a way that
P⊂E
1
2
3
E is complete and cocomplete
E has a model category structure in the sense of Quillen.
Exterior spaces are much easier to handle.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
An alternative category is that of the exterior spaces, introduced in
1998:
J.M. Garcia-Calcines, M. Garcia-Pinillos, L.J. Hernandez Paricio. A
closed simplicial model category for proper homotopy and shape
theories. Bull. Austral. Math. Soc., vol. 57 (1998), 221-242.
(Roughly speaking, an exterior space is a topological space with a
“neighborhood system at infinity")
The notion of exterior space is established in such a way that
P⊂E
1
2
3
E is complete and cocomplete
E has a model category structure in the sense of Quillen.
Exterior spaces are much easier to handle.
The category of exterior spaces has proved to be a useful framework for
proper homotopy theory.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Proper Whitehead Theorem
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Proper Whitehead Theorem
Proper Cellular Approximation Theorem
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Proper Whitehead Theorem
Proper Cellular Approximation Theorem
(they may also be proved within the proper setting)
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Proper Whitehead Theorem
Proper Cellular Approximation Theorem
(they may also be proved within the proper setting)
The Proper Blackers-Massey Theorem
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Proper Whitehead Theorem
Proper Cellular Approximation Theorem
(they may also be proved within the proper setting)
The Proper Blackers-Massey Theorem (unexpected in the general proper
setting, not necessarily cellular!).
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Proper Whitehead Theorem
Proper Cellular Approximation Theorem
(they may also be proved within the proper setting)
The Proper Blackers-Massey Theorem (unexpected in the general proper
setting, not necessarily cellular!).
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Proper Whitehead Theorem
Proper Cellular Approximation Theorem
(they may also be proved within the proper setting)
The Proper Blackers-Massey Theorem (unexpected in the general proper
setting, not necessarily cellular!).
Our strategy!
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Proper Whitehead Theorem
Proper Cellular Approximation Theorem
(they may also be proved within the proper setting)
The Proper Blackers-Massey Theorem (unexpected in the general proper
setting, not necessarily cellular!).
Our strategy!
Use the category E as a framework and work with exterior
CW-complexes
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Proper Whitehead Theorem
Proper Cellular Approximation Theorem
(they may also be proved within the proper setting)
The Proper Blackers-Massey Theorem (unexpected in the general proper
setting, not necessarily cellular!).
Our strategy!
Use the category E as a framework and work with exterior
CW-complexes
Obtain the theorems in the exterior setting (the proofs are analogous to
the ones found in the classical topological case)
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
In this work we do a deeper study of the category of exterior spaces by
developing a theory of exterior CW-complexes. This study will give
several interesting consequences in proper homotopy. Among these
results we can mention
Proper Whitehead Theorem
Proper Cellular Approximation Theorem
(they may also be proved within the proper setting)
The Proper Blackers-Massey Theorem (unexpected in the general proper
setting, not necessarily cellular!).
Our strategy!
Use the category E as a framework and work with exterior
CW-complexes
Obtain the theorems in the exterior setting (the proofs are analogous to
the ones found in the classical topological case)
Obtain the proper results when we restrict to P.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Proper and exterior homotopy.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Proper and exterior homotopy.
We begin by recalling notions concerning the proper category and the
category of exterior spaces.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Proper and exterior homotopy.
We begin by recalling notions concerning the proper category and the
category of exterior spaces.
f : X → Y is proper if it is continuous and f −1 (K) is compact (and
closed) for all K ⊂ Y closed compact subset.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Proper and exterior homotopy.
We begin by recalling notions concerning the proper category and the
category of exterior spaces.
f : X → Y is proper if it is continuous and f −1 (K) is compact (and
closed) for all K ⊂ Y closed compact subset.
We shall denote by P the category of spaces and proper maps.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Proper and exterior homotopy.
We begin by recalling notions concerning the proper category and the
category of exterior spaces.
f : X → Y is proper if it is continuous and f −1 (K) is compact (and
closed) for all K ⊂ Y closed compact subset.
We shall denote by P the category of spaces and proper maps. Proper
homotopy is defined in the natural way.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Proper and exterior homotopy.
We begin by recalling notions concerning the proper category and the
category of exterior spaces.
f : X → Y is proper if it is continuous and f −1 (K) is compact (and
closed) for all K ⊂ Y closed compact subset.
We shall denote by P the category of spaces and proper maps. Proper
homotopy is defined in the natural way.
R+ = [0, ∞).
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Proper and exterior homotopy.
We begin by recalling notions concerning the proper category and the
category of exterior spaces.
f : X → Y is proper if it is continuous and f −1 (K) is compact (and
closed) for all K ⊂ Y closed compact subset.
We shall denote by P the category of spaces and proper maps. Proper
homotopy is defined in the natural way.
R+ = [0, ∞).
A proper map α : R+ → X is called base ray in X. (We will also
consider base sequences, i.e. proper maps N → X)
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior spaces. Main properties.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior spaces. Main properties.
Definition (Pinillos-Calcines-Paricio, 1998)
An exterior space (X, ε ⊂ τ ) consists of a topological space (X, τ ) together
with a nonempty family of open subsets ε (called externology) satisfying
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior spaces. Main properties.
Definition (Pinillos-Calcines-Paricio, 1998)
An exterior space (X, ε ⊂ τ ) consists of a topological space (X, τ ) together
with a nonempty family of open subsets ε (called externology) satisfying
If E, E0 ∈ ε then E ∩ E0 ∈ ε
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior spaces. Main properties.
Definition (Pinillos-Calcines-Paricio, 1998)
An exterior space (X, ε ⊂ τ ) consists of a topological space (X, τ ) together
with a nonempty family of open subsets ε (called externology) satisfying
If E, E0 ∈ ε then E ∩ E0 ∈ ε
If E ∈ ε, U ∈ τ and E ⊂ U, then U ∈ ε.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior spaces. Main properties.
Definition (Pinillos-Calcines-Paricio, 1998)
An exterior space (X, ε ⊂ τ ) consists of a topological space (X, τ ) together
with a nonempty family of open subsets ε (called externology) satisfying
If E, E0 ∈ ε then E ∩ E0 ∈ ε
If E ∈ ε, U ∈ τ and E ⊂ U, then U ∈ ε.
A continuous map f : (X, ε ⊂ τ ) → (X 0 , ε0 ⊂ τ 0 ) is said to be exterior if
f −1 (E) ∈ ε for all E ∈ ε0 .
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior spaces. Main properties.
Definition (Pinillos-Calcines-Paricio, 1998)
An exterior space (X, ε ⊂ τ ) consists of a topological space (X, τ ) together
with a nonempty family of open subsets ε (called externology) satisfying
If E, E0 ∈ ε then E ∩ E0 ∈ ε
If E ∈ ε, U ∈ τ and E ⊂ U, then U ∈ ε.
A continuous map f : (X, ε ⊂ τ ) → (X 0 , ε0 ⊂ τ 0 ) is said to be exterior if
f −1 (E) ∈ ε for all E ∈ ε0 .
We will denote the category of exterior spaces by E.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior spaces. Main properties.
Definition (Pinillos-Calcines-Paricio, 1998)
An exterior space (X, ε ⊂ τ ) consists of a topological space (X, τ ) together
with a nonempty family of open subsets ε (called externology) satisfying
If E, E0 ∈ ε then E ∩ E0 ∈ ε
If E ∈ ε, U ∈ τ and E ⊂ U, then U ∈ ε.
A continuous map f : (X, ε ⊂ τ ) → (X 0 , ε0 ⊂ τ 0 ) is said to be exterior if
f −1 (E) ∈ ε for all E ∈ ε0 .
We will denote the category of exterior spaces by E.
Let X be a topological space. Then we can consider the cocompact
externology εcc = {X − K : K is closed and compact}. The
corresponding exterior space is denoted by Xcc . This construction gives
rise to a full embedding
(−)cc : P ,→ E
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior cylinder and exterior homotopy
Definition
Let X be any exterior space and let Y be any compact topological space. We
¯ as follows:
define the exterior space X ×Y
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior cylinder and exterior homotopy
Definition
Let X be any exterior space and let Y be any compact topological space. We
¯ as follows:
define the exterior space X ×Y
Its underlying topological space is the product X × Y
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior cylinder and exterior homotopy
Definition
Let X be any exterior space and let Y be any compact topological space. We
¯ as follows:
define the exterior space X ×Y
Its underlying topological space is the product X × Y
An open set E is exterior if there exists an exterior open subset G of X
for which G × Y ⊂ E.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior cylinder and exterior homotopy
Definition
Let X be any exterior space and let Y be any compact topological space. We
¯ as follows:
define the exterior space X ×Y
Its underlying topological space is the product X × Y
An open set E is exterior if there exists an exterior open subset G of X
for which G × Y ⊂ E.
Remarks:
¯ = (X × Y)cc
Xcc ×Y
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior cylinder and exterior homotopy
Definition
Let X be any exterior space and let Y be any compact topological space. We
¯ as follows:
define the exterior space X ×Y
Its underlying topological space is the product X × Y
An open set E is exterior if there exists an exterior open subset G of X
for which G × Y ⊂ E.
Remarks:
¯ = (X × Y)cc
Xcc ×Y
¯ Therefore, we can
We obtain the notion of exterior cylinder of X, X ×I.
define exterior homotopy (and exterior homotopy relative to R+ or N)
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior cylinder and exterior homotopy
Definition
Let X be any exterior space and let Y be any compact topological space. We
¯ as follows:
define the exterior space X ×Y
Its underlying topological space is the product X × Y
An open set E is exterior if there exists an exterior open subset G of X
for which G × Y ⊂ E.
Remarks:
¯ = (X × Y)cc
Xcc ×Y
¯ Therefore, we can
We obtain the notion of exterior cylinder of X, X ×I.
define exterior homotopy (and exterior homotopy relative to R+ or N)
We shall denote by [X, Y], [(X, α), (Y, β)]R+ , or [(X, α), (Y, β)]N the
corresponding homotopy brackets.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
The definition of exterior CW-complex
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
The definition of exterior CW-complex
Notation: N with the cocompact externology.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
The definition of exterior CW-complex
Notation: N with the cocompact externology.
¯ k
The k-dimensional N-sphere: Sk = N×S
k
¯ k
The k-dimensional N-disk:
D = N×D
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
The definition of exterior CW-complex
Notation: N with the cocompact externology.
¯ k
The k-dimensional N-sphere: Sk = N×S
k
¯ k
The k-dimensional N-disk:
D = N×D
Definition (Exterior CW-complex)
A relative exterior CW-complex (X, A) is an exterior space X together with a
filtration of exterior spaces A = X −1 ⊂ X 0 ⊂ X 1 ⊂ . . . ⊂ Xn ⊂ . . ., for
which X is its colimit and, for each n ≥ 0, X n is obtained from X n−1 as the
exterior pushout
tγ∈Γ Sn−1
_ γ
tγ∈Γ Dnγ
{ϕγ }γ∈Γ
{ψγ }γ∈Γ
via the attaching maps ϕγ : Sn−1 → X n−1 .
/ X n−1
_
/ Xn
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Examples
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Examples
Many classical objects studied in proper homotopy theory are easily checked
to be exterior CW-complexes considering their cocompact exterior structure:
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Examples
Many classical objects studied in proper homotopy theory are easily checked
to be exterior CW-complexes considering their cocompact exterior structure:
The Brown sphere SBn , has an exterior CW-decomposition in which
R+ = X −1 = X 0 = . . . = X n−1 and X n = SBn is obtained by attaching
an N-cell via ϕ : Sn−1 → R+ , (n, x) 7→ n.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Examples
Many classical objects studied in proper homotopy theory are easily checked
to be exterior CW-complexes considering their cocompact exterior structure:
The Brown sphere SBn , has an exterior CW-decomposition in which
R+ = X −1 = X 0 = . . . = X n−1 and X n = SBn is obtained by attaching
an N-cell via ϕ : Sn−1 → R+ , (n, x) 7→ n.
Open differential manifolds and PL-manifolds are exterior
CW-complexes as they admit a locally finite countable triangulation
which describes the exterior CW-structure.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Examples
Many classical objects studied in proper homotopy theory are easily checked
to be exterior CW-complexes considering their cocompact exterior structure:
The Brown sphere SBn , has an exterior CW-decomposition in which
R+ = X −1 = X 0 = . . . = X n−1 and X n = SBn is obtained by attaching
an N-cell via ϕ : Sn−1 → R+ , (n, x) 7→ n.
Open differential manifolds and PL-manifolds are exterior
CW-complexes as they admit a locally finite countable triangulation
which describes the exterior CW-structure.
Non compact finite dimensional locally finite polyhedra, in particular
open topological manifolds of dimension 2 and 3, are exterior
CW-complexes.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Examples
Many classical objects studied in proper homotopy theory are easily checked
to be exterior CW-complexes considering their cocompact exterior structure:
The Brown sphere SBn , has an exterior CW-decomposition in which
R+ = X −1 = X 0 = . . . = X n−1 and X n = SBn is obtained by attaching
an N-cell via ϕ : Sn−1 → R+ , (n, x) 7→ n.
Open differential manifolds and PL-manifolds are exterior
CW-complexes as they admit a locally finite countable triangulation
which describes the exterior CW-structure.
Non compact finite dimensional locally finite polyhedra, in particular
open topological manifolds of dimension 2 and 3, are exterior
CW-complexes.
Given an exterior CW-complex X and a classical finite CW-complex K
¯ admits an exterior CW-structure.
of dimension m, X ×K
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Examples
Many classical objects studied in proper homotopy theory are easily checked
to be exterior CW-complexes considering their cocompact exterior structure:
The Brown sphere SBn , has an exterior CW-decomposition in which
R+ = X −1 = X 0 = . . . = X n−1 and X n = SBn is obtained by attaching
an N-cell via ϕ : Sn−1 → R+ , (n, x) 7→ n.
Open differential manifolds and PL-manifolds are exterior
CW-complexes as they admit a locally finite countable triangulation
which describes the exterior CW-structure.
Non compact finite dimensional locally finite polyhedra, in particular
open topological manifolds of dimension 2 and 3, are exterior
CW-complexes.
Given an exterior CW-complex X and a classical finite CW-complex K
¯ admits an exterior CW-structure.
of dimension m, X ×K
Let (X, A) be a locally finite, finite dimensional relative CW-complex in
which, for any k, X has no k-cells or has infinite countable many
k-cells. Then (Xcc , Acc ) is an exterior relative CW-complex.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior homotopy groups
We consider N-spheres and N-discs as objects in EN via the base sequence
η : N → Sk ⊂ Dk+1 ,
η(n) = (n, ∗).
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior homotopy groups
We consider N-spheres and N-discs as objects in EN via the base sequence
η : N → Sk ⊂ Dk+1 ,
¯ k;
Remember: Sk = N×S
¯ k
Dk = N×D
η(n) = (n, ∗).
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior homotopy groups
We consider N-spheres and N-discs as objects in EN via the base sequence
η : N → Sk ⊂ Dk+1 ,
¯ k;
Remember: Sk = N×S
η(n) = (n, ∗).
¯ k
Dk = N×D
Definition
The k-th Brown-Grossman exterior homotopy group of (X, α) ∈ ER+ , k ≥ 0,
is defined as πkB (X, α) = [(Sk , η), (X, α|N )]N .
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior homotopy groups
We consider N-spheres and N-discs as objects in EN via the base sequence
η : N → Sk ⊂ Dk+1 ,
¯ k;
Remember: Sk = N×S
η(n) = (n, ∗).
¯ k
Dk = N×D
Definition
The k-th Brown-Grossman exterior homotopy group of (X, α) ∈ ER+ , k ≥ 0,
is defined as πkB (X, α) = [(Sk , η), (X, α|N )]N .
Remarks:
We can also consider exterior homotopy groups for (X, α) ∈ EN
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior homotopy groups
We consider N-spheres and N-discs as objects in EN via the base sequence
η : N → Sk ⊂ Dk+1 ,
¯ k;
Remember: Sk = N×S
η(n) = (n, ∗).
¯ k
Dk = N×D
Definition
The k-th Brown-Grossman exterior homotopy group of (X, α) ∈ ER+ , k ≥ 0,
is defined as πkB (X, α) = [(Sk , η), (X, α|N )]N .
Remarks:
We can also consider exterior homotopy groups for (X, α) ∈ EN
We can easily generalize to homotopy groups
πkB (X, A, α) = [(Dk , Sk−1 , η), (X, A, α)]N
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Exterior homotopy groups
We consider N-spheres and N-discs as objects in EN via the base sequence
η : N → Sk ⊂ Dk+1 ,
¯ k;
Remember: Sk = N×S
η(n) = (n, ∗).
¯ k
Dk = N×D
Definition
The k-th Brown-Grossman exterior homotopy group of (X, α) ∈ ER+ , k ≥ 0,
is defined as πkB (X, α) = [(Sk , η), (X, α|N )]N .
Remarks:
We can also consider exterior homotopy groups for (X, α) ∈ EN
We can easily generalize to homotopy groups
πkB (X, A, α) = [(Dk , Sk−1 , η), (X, A, α)]N
In the proper homotopy setting, i.e., whenever (X, α) ∈ PR+ , the
exterior homotopy groups of (Xcc , αcc ) are the (global)
Brown-Grossmann proper homotopy groups.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Equivalences
Definition
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Equivalences
Definition
An exterior map f : Y → Z is an exterior n-equivalence or simply
e-n-equivalence if for any exterior base sequence α : N → Y
∼
=
f∗ : πkB (Y, α) −→ πkB (Z, f α) is isomorphism for 0 ≤ k ≤ n − 1
and f∗ : πnB (Y, α) −→ πnB (Z, f α) is surjective
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Equivalences
Definition
An exterior map f : Y → Z is an exterior n-equivalence or simply
e-n-equivalence if for any exterior base sequence α : N → Y
∼
=
f∗ : πkB (Y, α) −→ πkB (Z, f α) is isomorphism for 0 ≤ k ≤ n − 1
and f∗ : πnB (Y, α) −→ πnB (Z, f α) is surjective
An exterior space X ∈ E is e-n-connected if, for any exterior base
sequence α : N → X,
πkB (X, α) = {0}, 0 ≤ k ≤ n
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Equivalences
Definition
An exterior map f : Y → Z is an exterior n-equivalence or simply
e-n-equivalence if for any exterior base sequence α : N → Y
∼
=
f∗ : πkB (Y, α) −→ πkB (Z, f α) is isomorphism for 0 ≤ k ≤ n − 1
and f∗ : πnB (Y, α) −→ πnB (Z, f α) is surjective
An exterior space X ∈ E is e-n-connected if, for any exterior base
sequence α : N → X,
πkB (X, α) = {0}, 0 ≤ k ≤ n
Remarks:
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Equivalences
Definition
An exterior map f : Y → Z is an exterior n-equivalence or simply
e-n-equivalence if for any exterior base sequence α : N → Y
∼
=
f∗ : πkB (Y, α) −→ πkB (Z, f α) is isomorphism for 0 ≤ k ≤ n − 1
and f∗ : πnB (Y, α) −→ πnB (Z, f α) is surjective
An exterior space X ∈ E is e-n-connected if, for any exterior base
sequence α : N → X,
πkB (X, α) = {0}, 0 ≤ k ≤ n
Remarks:
We can establish: (X, A) e-n-connected;
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Equivalences
Definition
An exterior map f : Y → Z is an exterior n-equivalence or simply
e-n-equivalence if for any exterior base sequence α : N → Y
∼
=
f∗ : πkB (Y, α) −→ πkB (Z, f α) is isomorphism for 0 ≤ k ≤ n − 1
and f∗ : πnB (Y, α) −→ πnB (Z, f α) is surjective
An exterior space X ∈ E is e-n-connected if, for any exterior base
sequence α : N → X,
πkB (X, α) = {0}, 0 ≤ k ≤ n
Remarks:
We can establish: (X, A) e-n-connected;
p-n-equivalences (p-n-connected)
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Equivalences
Definition
An exterior map f : Y → Z is an exterior n-equivalence or simply
e-n-equivalence if for any exterior base sequence α : N → Y
∼
=
f∗ : πkB (Y, α) −→ πkB (Z, f α) is isomorphism for 0 ≤ k ≤ n − 1
and f∗ : πnB (Y, α) −→ πnB (Z, f α) is surjective
An exterior space X ∈ E is e-n-connected if, for any exterior base
sequence α : N → X,
πkB (X, α) = {0}, 0 ≤ k ≤ n
Remarks:
We can establish: (X, A) e-n-connected;
p-n-equivalences (p-n-connected)
In the proper setting, a space X ∈ P∞ is e-0-connected iff it is
one-ended.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Main theorems
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Main theorems
Ordinary homotopy theory of CW-complexes can be translated to E
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Main theorems
Ordinary homotopy theory of CW-complexes can be translated to E
Theorem (Exterior Whitehead)
Let f : (X, R+ ) → (Y, R+ ) be an e-n-equivalence (n ≤ ∞) between relative
exterior CW-complexes of dimension at most n. Then, f is an exterior
homotopy equivalence rel. R+ .
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Main theorems
Ordinary homotopy theory of CW-complexes can be translated to E
Theorem (Exterior Whitehead)
Let f : (X, R+ ) → (Y, R+ ) be an e-n-equivalence (n ≤ ∞) between relative
exterior CW-complexes of dimension at most n. Then, f is an exterior
homotopy equivalence rel. R+ .
Theorem (Exterior Cellular Approximation)
Given an exterior map f : (X, A) → (Y, B) between exterior relative
CW-complexes, there exists and exterior and cellular map
g : (X, A) → (Y, B) for which g ' f rel. A.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Main theorems
Ordinary homotopy theory of CW-complexes can be translated to E
Theorem (Exterior Whitehead)
Let f : (X, R+ ) → (Y, R+ ) be an e-n-equivalence (n ≤ ∞) between relative
exterior CW-complexes of dimension at most n. Then, f is an exterior
homotopy equivalence rel. R+ .
Theorem (Exterior Cellular Approximation)
Given an exterior map f : (X, A) → (Y, B) between exterior relative
CW-complexes, there exists and exterior and cellular map
g : (X, A) → (Y, B) for which g ' f rel. A.
Theorem (Exterior CW-approximation)
Given an e-0-connected space (X, α) ∈ ER+ there exists a exterior
∞-equivalence
∼
b→
X
w:X
b R+ ) is a relative exterior CW-complex.
in which (X,
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Main theorems
Theorem (Exterior Blakers-Massey)
Let X = X1 ∪ X2 , A = X1 ∩ X2 in which (X1 , A) and (X2 , A) are exterior
cofibred pairs; X1 , X2 and A are e-0-connected; (X1 , A) is
e-(n − 1)-connected, and (X2 , A) is e-(m − 1)-connected, m, n ≥ 1. Then
(X1 , A) 
is an e-(m + n − 2)-equivalence.
/ (X, X2 )
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Main theorems
Theorem (Exterior Blakers-Massey)
Let X = X1 ∪ X2 , A = X1 ∩ X2 in which (X1 , A) and (X2 , A) are exterior
cofibred pairs; X1 , X2 and A are e-0-connected; (X1 , A) is
e-(n − 1)-connected, and (X2 , A) is e-(m − 1)-connected, m, n ≥ 1. Then
(X1 , A) 
/ (X, X2 )
is an e-(m + n − 2)-equivalence.
An exterior map is an exterior cofibration if it satisfies the usual
homotopy extension property (HEP) in E. An exterior pair (X, A) is
cofibred provided the inclusion is a (closed) exterior cofibration.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Main theorems
Theorem (Exterior Blakers-Massey)
Let X = X1 ∪ X2 , A = X1 ∩ X2 in which (X1 , A) and (X2 , A) are exterior
cofibred pairs; X1 , X2 and A are e-0-connected; (X1 , A) is
e-(n − 1)-connected, and (X2 , A) is e-(m − 1)-connected, m, n ≥ 1. Then
(X1 , A) 
/ (X, X2 )
is an e-(m + n − 2)-equivalence.
An exterior map is an exterior cofibration if it satisfies the usual
homotopy extension property (HEP) in E. An exterior pair (X, A) is
cofibred provided the inclusion is a (closed) exterior cofibration.
Given a proper map j : A → X between Hausdorff locally compact
spaces, then j is a proper cofibration if and only if jcc is an exterior
cofibration.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Theorem (Proper Whitehead)
If f : (X, R+ ) → (Y, R+ ) is a p-n-equivalence between locally finite
CW-complexes with finite dimension less than n and for each 0 ≤ k ≤ d
either X (respec. Y) has no k-cells or X (respec. Y) has an infinite countable
number of k-cells, then f is a proper homotopy equivalence rel. R+ .
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Theorem (Proper Whitehead)
If f : (X, R+ ) → (Y, R+ ) is a p-n-equivalence between locally finite
CW-complexes with finite dimension less than n and for each 0 ≤ k ≤ d
either X (respec. Y) has no k-cells or X (respec. Y) has an infinite countable
number of k-cells, then f is a proper homotopy equivalence rel. R+ .
Theorem (Proper Cellular Approximation)
If f : (X, R+ ) → (Y, R+ ) is a p-n-equivalence between locally finite
CW-complexes with finite dimension less than n and for each 0 ≤ k ≤ d
either X (respec. Y) has no k-cells or X (respec. Y) has an infinite countable
number of k-cells, then there exists a cellular proper map
g : (X, A) → (Y, B) with g 'p f rel. A.
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Theorem (Proper Whitehead)
If f : (X, R+ ) → (Y, R+ ) is a p-n-equivalence between locally finite
CW-complexes with finite dimension less than n and for each 0 ≤ k ≤ d
either X (respec. Y) has no k-cells or X (respec. Y) has an infinite countable
number of k-cells, then f is a proper homotopy equivalence rel. R+ .
Theorem (Proper Cellular Approximation)
If f : (X, R+ ) → (Y, R+ ) is a p-n-equivalence between locally finite
CW-complexes with finite dimension less than n and for each 0 ≤ k ≤ d
either X (respec. Y) has no k-cells or X (respec. Y) has an infinite countable
number of k-cells, then there exists a cellular proper map
g : (X, A) → (Y, B) with g 'p f rel. A.
Remark: There is no a proper version of a CW-approximation theorem. If
(X, α) ∈ PR+ (X one-ended) the construction of the relative exterior
b R+ ) gives a non cocompact exterior space.
CW-complex (X,
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Theorem (Proper Blakers-Massey)
Given X = X1 ∪ X2 , A = X1 ∩ X2 with (X1 , A) and (X2 , A) proper cofibred
R
pairs and X1 , X2 and A one-ended spaces in Pw + . If (X1 , A) is
p-(n − 1)-conected and (X2 , A) is p-(m − 1)-conected (m, n ≥ 1), then
(X1 , A) is a p-(m + n − 2)-equivalence.

/ (X, X2 )
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Theorem (Proper Blakers-Massey)
Given X = X1 ∪ X2 , A = X1 ∩ X2 with (X1 , A) and (X2 , A) proper cofibred
R
pairs and X1 , X2 and A one-ended spaces in Pw + . If (X1 , A) is
p-(n − 1)-conected and (X2 , A) is p-(m − 1)-conected (m, n ≥ 1), then
(X1 , A) is a p-(m + n − 2)-equivalence.
Sketch of the proof:

/ (X, X2 )
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Theorem (Proper Blakers-Massey)
Given X = X1 ∪ X2 , A = X1 ∩ X2 with (X1 , A) and (X2 , A) proper cofibred
R
pairs and X1 , X2 and A one-ended spaces in Pw + . If (X1 , A) is
p-(n − 1)-conected and (X2 , A) is p-(m − 1)-conected (m, n ≥ 1), then
(X1 , A) 
/ (X, X2 )
is a p-(m + n − 2)-equivalence.
Sketch of the proof:
Consider cocompact externologies and work in the exterior setting
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Theorem (Proper Blakers-Massey)
Given X = X1 ∪ X2 , A = X1 ∩ X2 with (X1 , A) and (X2 , A) proper cofibred
R
pairs and X1 , X2 and A one-ended spaces in Pw + . If (X1 , A) is
p-(n − 1)-conected and (X2 , A) is p-(m − 1)-conected (m, n ≥ 1), then
(X1 , A) 
/ (X, X2 )
is a p-(m + n − 2)-equivalence.
Sketch of the proof:
Consider cocompact externologies and work in the exterior setting
Consider exterior CW-approximations (do not have to be cocompact!)
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Theorem (Proper Blakers-Massey)
Given X = X1 ∪ X2 , A = X1 ∩ X2 with (X1 , A) and (X2 , A) proper cofibred
R
pairs and X1 , X2 and A one-ended spaces in Pw + . If (X1 , A) is
p-(n − 1)-conected and (X2 , A) is p-(m − 1)-conected (m, n ≥ 1), then
(X1 , A) 
/ (X, X2 )
is a p-(m + n − 2)-equivalence.
Sketch of the proof:
Consider cocompact externologies and work in the exterior setting
Consider exterior CW-approximations (do not have to be cocompact!)
Prove the result (similar to the classical proof)
Introduction. Preliminary notions in proper homotopy. The category of exterior spaces. Exterior CW-complexes. The main theorems for exterior CW-complexes Conseque
Consequences in proper homotopy theory
Theorem (Proper Blakers-Massey)
Given X = X1 ∪ X2 , A = X1 ∩ X2 with (X1 , A) and (X2 , A) proper cofibred
R
pairs and X1 , X2 and A one-ended spaces in Pw + . If (X1 , A) is
p-(n − 1)-conected and (X2 , A) is p-(m − 1)-conected (m, n ≥ 1), then
(X1 , A) 
/ (X, X2 )
is a p-(m + n − 2)-equivalence.
Sketch of the proof:
Consider cocompact externologies and work in the exterior setting
Consider exterior CW-approximations (do not have to be cocompact!)
Prove the result (similar to the classical proof)
THANKS FOR LISTENING!