Download Algebraic Topology of Finite Topological Spaces and by Jonathan A. Barmak 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 point of view. In particular,
the equipment constructed during this manuscript are used to check Quillen’s
conjecture at the poset of p-subgroups of a finite crew 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 urged 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

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The finite T0 -space X (K) associated to K (also called the face poset of K) is the poset of simplices of K ordered by inclusion. If ϕ : K → L is a simplicial map between finite simplicial complexes, there is a continuous map X (ϕ) : X (K) → X (L) defined by X (ϕ)(σ) = ϕ(σ) for every simplex σ of K. 11. If K is the 2-simplex, the associated finite space is the following • Ò ``` Ò `` Ò `` ÒÒ ÒÒ • • • `` Ò `` Ò ``ÒÒ ``ÒÒ ÒÒ`` ÒÒ`` ÒÒ ` ÒÒ ` • • • If K is a finite complex, K(X (K)) is the first barycentric subdivision K of K and if ϕ : K → L is a simplicial map, K(X (ϕ)) = ϕ : K → L is the map induced in the barycentric subdivisions.

Then (Y, A) is a minimal pair. 3. Therefore A and Y are homeomorphic and so, X Y = A. 6. The space X • • • XX Ô XXÔÔ ÔÔXX ÔÔ X XX Ô XXÔÔ ÔÔXX Ô Ô X • aa aa aa a • x aa Ñ • aaÑÑ ÑÑaaa Ñ Ñ • • is contractible, but the point x is not a strong deformation retract of X, because (X, {x}) is a minimal pair. 7. Let (X, A) be a minimal pair such that A is a minimal finite space and f 1(X,A) : (X, A) → (X, A). Then f = 1X . If X and Y are homotopy equivalent finite T0 -spaces, the associated polyhedra |K(X)| and |K(Y )| also have the same homotopy type.

In 1946 Birkhoff [13] proved that if the order of G is n, G can be realized as the automorphisms of a poset with n(n + 1) points. In 1972 Thornton [78] improved slightly Birkhoff’s result: He obtained a poset of n(2r + 1) points, when the group is generated by r elements. We present here a result which appears in [10]. 1. Given a group G of finite order n with r generators, there exists a poset X with n(r + 2) points such that Aut(X) G. Recall first that the height ht(X) of a finite poset X is one less than the maximum number of elements in a chain of X.

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