Download Algebraic Topology of Finite Topological Spaces and by Jonathan A. Barmak PDF

By Jonathan A. Barmak

This quantity offers with the idea of finite topological areas and its
relationship with the homotopy and easy homotopy idea of polyhedra.
The interplay among their intrinsic combinatorial and topological
structures makes finite areas a useful gizmo for learning difficulties in
Topology, Algebra and Geometry from a brand new point of view. In particular,
the equipment constructed during this manuscript are used to review Quillen’s
conjecture at the poset of p-subgroups of a finite team and the
Andrews-Curtis conjecture at the 3-deformability of contractible
two-dimensional complexes.
This self-contained paintings constitutes the 1st detailed
exposition at the algebraic topology of finite areas. it truly is intended
for topologists and combinatorialists, however it is usually steered for
advanced undergraduate scholars and graduate scholars with a modest
knowledge of Algebraic Topology.

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Additional resources for Algebraic Topology of Finite Topological Spaces and Applications

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Then X tractible if and only if X or Y is contractible. Y is con- Proof. Assume X is contractible. Then there exists a sequence of spaces X = Xn Xn−1 ... X1 = {x1 } with Xi = {x1 , x2 , . . , xi } and such that xi is a beat point of Xi for every 2 ≤ i ≤ n. Then xi is a beat point of Xi Y for each 2 ≤ i ≤ n and therefore, X Y deformation retracts to {x1 } Y which is contractible. Analogously, if Y is contractible, so is X Y . Now suppose X Y is contractible. Then there exists a sequence X Y = Xn Yn with Xi ⊆ X, Yi ⊆ Y , Xi of Xi Yi for i ≥ 2.

Proof. Let S = {σ1 , σ2 , . . , σr } be a simplex of K , where σ1 σr is a chain of simplices of K. Let α be a point in the closed simplex S. Then sK (α) ∈ σr ⊆ |K| and |ϕ|sK (α) ∈ ϕ(σr ) ⊆ |L|. On the other hand, |ϕ |(α) ∈ {ϕ(σ1 ), ϕ(σ2 ), . . , ϕ(σr )} and then sL |ϕ |(α) ∈ ϕ(σr ). Therefore, the linear homotopy H : |K | × I → |L|, (α, t) → (1 − t)|ϕ|sK (α) + tsL |ϕ |(α) is well defined and continuous. 7, μL |ϕ| = μX (L) s−1 L |ϕ| μX (L) |ϕ |s−1 K = X (ϕ)μX (K) s−1 K = X (ϕ)μK . 14. An explicit homotopy between μL |ϕ| and X (ϕ)μK is H = μL H(s−1 K × 1I ).

We claim that μ−1 |L|. X (Ux ) = |K(X)| If α ∈ μ−1 X (Ux ), then min(support(α)) ∈ Ux . In particular, the support of α contains a vertex of Ux and then α ∈ / |L|. Conversely, if α ∈ / |L|, there exists y ∈ support(α) such that y ∈ Ux . Then min(support(α)) ≤ y ≤ x and therefore μX (α) ∈ Ux . Since |L| ⊆ |K(X)| is closed, μ−1 X (Ux ) is open. Now we show that |K(Ux )| is a strong deformation retract of |K(X)| |L|. This is a particular case of a more general fact. Let i : |K(Ux )| → |K(X)| |L| be the inclusion.

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