Jonathan A. Barmak's Algebraic Topology of Finite Topological Spaces and PDF

By Jonathan A. Barmak

This quantity bargains with the speculation of finite topological areas and its
relationship with the homotopy and straightforward 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 check Quillen’s
conjecture at the poset of p-subgroups of a finite staff 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 additionally steered for
advanced undergraduate scholars and graduate scholars with a modest
knowledge of Algebraic Topology.

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Extra info for Algebraic Topology of Finite Topological Spaces and Applications

Example text

Now, for each 1 ≤ k ≤ n attach a chain of length kn to X with minimum xn−k+1 . The resulting space ˜ deformation retracts to X and every automorphism f : X ˜ →X ˜ must fix X 2 the unique chain C1 of length n (with minimum x1 ). Therefore f restricts ˜ C1 → X ˜ C1 which must fix the unique chain C2 to a homeomorphism X ˜ of length n(n − 1) of X C1 (with minimum x2 ). Applying this reasoning ˜ On the other hand, we repeatedly, we conclude that f fixes every point of X. know that there exists a finite T0 -space Z such that Aut(Z) = G.

Then the restrictions f |Ux , φ(g)|Ux : Ux → Uf (x) are isomorphisms. On the other hand, there exists a unique automorphism Ux → Ux since the unique chain of i + 2 elements must be fixed by any such automorphism. Thus, f |−1 Ux φ(g)|Ux = 1Ux , and then f |Ux = φ(g)|Ux , which proves that Ux ⊆ Y . Similarly we see that Y ⊆ X is closed. Assume x = (h, i) ∈ / Y . Since f ∈ Aut(X), it preserves the height of any point. In particular ht(f (x)) = ht(x) = i + 1 and therefore f (x) = (k, i) = φ(kh−1 )(x) for some k ∈ G.

1. Let f, g : X → Y be two homotopic maps between finite T0 spaces. Then there exists a sequence f = f0 , f1 , . . , fn = g such that for every 0 ≤ i < n there is a point xi ∈ X with the following properties: 1. fi and fi+1 coincide in X {xi }, and 2. fi (xi ) ≺ fi+1 (xi ) or fi+1 (xi ) ≺ fi (xi ). Proof. 6. Let A = {x ∈ X | f (x) = g(x)}. If A = ∅, f = g and there is nothing to prove. A. 1007/978-3-642-22003-6 2, © Springer-Verlag Berlin Heidelberg 2011 19 20 3 Basic Topological Properties of Finite Spaces of A.

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