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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!