By Jonathan A. Barmak

This quantity bargains with the speculation 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 great tool for learning difficulties in

Topology, Algebra and Geometry from a brand new viewpoint. In particular,

the equipment constructed during this manuscript are used to check 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 can be steered for

advanced undergraduate scholars and graduate scholars with a modest

knowledge of Algebraic Topology.

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**Extra resources for Algebraic Topology of Finite Topological Spaces and Applications **

**Sample text**

I=0 (xk , yk ) is a chain in X × Y , f (α) = ( k ti xi , i=0 k ti yi ). i=0 Since |K(pX )| and |K(pY )| are continuous, so is f . |K(X × Y )| is compact and |K(X)|×|K(Y )| is Hausdorﬀ, so we only need to show that f is a bijection. Details will be left to the reader. An explicit formula for g = f −1 is given by k l u i xi , g( i=0 vi yi ) = i=0 tij (xi , yj ), i,j 32 3 Basic Topological Properties of Finite Spaces where tij = max{0, min{u0 + u1 + . . + ui , v0 + v1 + . . vj } − max{u0 + u1 + .

Let f : X → Y be an order preserving map between finite posets such that |K(f −1 (Uy ))| is contractible for every y ∈ Y . Then |K(f )| : |K(X)| → |K(Y )| is a homotopy equivalence. 20. In Quillen’s paper [70] and in many other articles ([17, 18, 79, 80] for instance), posets are studied from a topological viewpoint only through their associated simplicial complexes. In some of those papers, when it is said that a ﬁnite poset is contractible it is meant that the associated polyhedron is contractible, and when an order preserving map is claimed to be a homotopy equivalence, this is regarded as the simplicial map between the associated complexes.

E1 . If ξ, ξ are H-paths such that e(ξ) = o(ξ ), we deﬁne the product H-path ξξ as the concatenation of the sequence ξ followed by the sequence ξ . An H-path ξ = e1 e2 . . en is said to be monotonic if ei ∈ E(H(X)) for all 1 ≤ i ≤ n or e−1 ∈ E(H(X)) for all 1 ≤ i ≤ n. i A loop at x0 is an H-path that starts and ends in x0 . Given two loops ξ, ξ at x0 , we say that they are close if there exist H-paths ξ1 , ξ2 , ξ3 , ξ4 such that ξ2 and ξ3 are monotonic and the set {ξ, ξ } coincides with {ξ1 ξ2 ξ3 ξ4 , ξ1 ξ4 }.