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

By Jonathan A. Barmak

This quantity bargains with the speculation of finite topological areas and its
relationship with the homotopy and easy homotopy concept 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 viewpoint. In particular,
the tools built during this manuscript are used to review Quillen’s
conjecture at the poset of p-subgroups of a finite workforce 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 can also be advised for
advanced undergraduate scholars and graduate scholars with a modest
knowledge of Algebraic Topology.

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Example text

Define the K-McCord map μX : |K(X)| → X by μX (α) = min(support(α)). 1. 6. The K-McCord map μX is a weak homotopy equivalence for every finite T0 -space X. Proof. Notice that the minimal open sets Ux are contractible because they have maximum. We will prove that for each x ∈ X, μ−1 X (Ux ) is open and contractible. This will show that μX is continuous and that the restrictions μX |μ−1 (Ux ) : μ−1 X (Ux ) → Ux are weak homotopy equivalences. 2, μX is a weak homotopy equivalence. 4 Weak Homotopy Types: The Theory of McCord 13 Let x ∈ X and let L = K(X Ux ) ⊆ K(X).

All Hausdorff spaces are T1 , but the converse is false. If a finite space is T1 , then every subset is closed and so, X is discrete. 1. Let X be a finite space such that Xc is discrete. Then X is a disjoint union of contractible spaces. 2. Let X be a finite space and let Y be a T1 -space homotopy equivalent to X. Then X is a disjoint union of contractible spaces. Proof. Since X Y , Xc Y . Let f : Xc → Y be a homotopy equivalence with homotopy inverse g. 6. Since f is a one to one map from Xc to a T1 -space, it follows that Xc is also T1 and therefore discrete.

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. Xn−1 Yn−1 ... X1 Y1 = {z1 } Yi = {z1 , z2 . . 7 Joins, Products, Quotients and Wedges 31 Let i ≥ 2. If zi ∈ Xi , zi is a beat point of Xi unless it is a maximal point of Xi and Yi has a minimum. In the same way, if zi ∈ Yi , zi is a beat point of Yi or Xi has a maximum.

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