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 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
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 prompt 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
Suppose that f ≤ 1X and f |A = 1A . Let x ∈ X. If x ∈ X is minimal, f (x) = x. In general, suppose we have proved that f |Uˆx = 1|Uˆx . If x ∈ A, f (x) = x. If x ∈ / A, x is not a down beat point of X. However y < x implies y = f (y) ≤ f (x) ≤ x. Therefore f (x) = x. 6. 3. Let (X, A) and (Y, B) be minimal pairs, f : X → Y , g : Y → X such that gf 1X rel A, gf 1Y rel B. Then f and g are homeomorphisms. 4. If x is a beat point of a ﬁnite T0 -space X, we say that there is an elementary strong collapse from X to X x and write X e X x.
2. 21. Let f : X → Y be a map between finite T0 -spaces such that f −1 (c) ⊆ X is homotopically trivial for every chain c of Y . Then f is a weak homotopy equivalence. Proof. If c is a chain of Y or, equivalently, a simplex of K(Y ), then |K(f )|−1 (c) = |K(f −1 (c))|, which is contractible since f −1 (c) is homotopically trivial. By Theorem A, |K(f )| is a homotopy equivalence and then f is a weak homotopy equivalence. 21 holds, then f −1 (Uy ) is homotopically trivial for every y ∈ Y and, by McCord Theorem, f is a weak homotopy equivalence.
2, f is a weak homotopy equivalence. However, f is not a homotopy equivalence since its source and target are non homeomorphic minimal spaces. 4. Let X be a ﬁnite T0 -space. The simplicial complex K(X) associated to X (also called the order complex ) is the simplicial complex whose simplices are the nonempty chains of X (see Fig. 3). Moreover, if f : X → Y is a continuous map between ﬁnite T0 -spaces, the associated simplicial map K(f ) : K(X) → K(Y ) is deﬁned by K(f )(x) = f (x). Note that if f : X → Y is a continuous map between ﬁnite T0 -spaces, the vertex map K(f ) : K(X) → K(Y ) is simplicial since f is order preserving and maps chains to chains.