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Über dieses Buch

This volume deals with the theory of finite topological spaces and its relationship with the homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic combinatorial and topological structures makes finite spaces a useful tool for studying problems in Topology, Algebra and Geometry from a new perspective. In particular, the methods developed in this manuscript are used to study Quillen's conjecture on the poset of p-subgroups of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible two-dimensional complexes. This self-contained work constitutes the first detailed exposition on the algebraic topology of finite spaces. It is intended for topologists and combinatorialists, but it is also recommended for advanced undergraduate students and graduate students with a modest knowledge of Algebraic Topology.



Chapter 1. Preliminaries

In this chapter we will state some of the most important results on finite spaces which are previous to our work. These results can be summarized by the following three items: 1. The correspondence between finite topological spaces and finite partially ordered sets, first considered by Alexandroff in [1] in 1937. The combinatorial description of homotopy types of finite spaces, discovered by Stong in his beautiful article [76] of 1966. 3. The connection between finite spaces and polyhedra, found by McCord [55] also in 1966.

Jonathan A. Barmak

Chapter 2. Basic Topological Properties of Finite Spaces

In this chapter we present some results concerning elementary topological aspects of finite spaces. The proofs use basic elements of Algebraic Topology and have a strong combinatorial flavour. We study further homotopical properties including classical homotopy invariants and finite analogues of well-known topological constructions.

Jonathan A. Barmak

Chapter 3. Minimal Finite Models

In Sect. 3.3 we proved that in general, if K is a finite simplicial complex, there is no finite space with the homotopy type of |K|. However, by Theorem 2.4.12 any compact polyhedron is weak homotopy equivalent to a finite space. In this chapter we will study finite models of polyhedra in this sense and we will describe the minimal finite models of some well-known (Hausdorff) spaces, i.e. weak homotopy equivalent finite spaces of minimum cardinality. The main results of this chapter appear in [7].

Jonathan A. Barmak

Chapter 4. Simple Homotopy Types and Finite Spaces

Whitehead’s theory of simple homotopy types is inspired by Tietze’s theorem in combinatorial group theory, which states that any finite presentation of a group could be deformed into any other by a finite sequence of elementary moves, which are now called Tietze transformations. Whitehead translated these algebraic moves into the well-known geometric moves of elementary collapses and expansions of finite simplicial complexes.

Jonathan A. Barmak

Chapter 5. Strong Homotopy Types

The notion of collapse of finite spaces is directly connected with the concept of simplicial collapse. In Chap. 3 we studied the notion of elementary strong collapse which is the fundamental move that describes homotopy types of finite spaces. In this chapter we will define the notion of strong collapse of simplicial complexes which leades to strong homotopy types of complexes.

Jonathan A. Barmak

Chapter 6. Methods of Reduction

A method of reduction of finite spaces is a technique that allows one to reduce the number of points of a finite topological space preserving some properties of the space.

Jonathan A. Barmak

Chapter 7. h-Regular Complexes and Quotients

The results of McCord show that each compact polyhedron |K| can be modeled, up to weak homotopy, by a finite space X(K). It is not hard to prove that this result can be extended to the so called regular CW-complexes. In this chapter we introduce a new class of complexes, generalizing the notion of simplicial complex and of regular complex, and we prove that they also can be modeled by their face posets.

Jonathan A. Barmak

Chapter 8. Group Actions and a Conjecture of Quillen

In his seminal article [70], Daniel Quillen studied algebraic properties of a finite group by means of homotopy properties of a certain compled K(S p (G)) associated to the group. Given a finite group G and a prime integer p dividing the order of G, let S p (G) denote the poset of nontrivial p-subgroups of G ordered by inclusion.

Jonathan A. Barmak

Chapter 9. Reduced Lattices

Recall that a poset P is said to be a lattice if every two-point set {a, b} has a least upper bound a ∨ b, called join or supremum of a and b, and a greatest lower bound a ∧ b, called meet or infimum. Any finite lattice has a maximum (and a minimum), and in particular it is a contractible finite space. In this chapter we will study the spaces obtained from a lattice by removing its maximum and its minimum, which are more attractive from a topological point of view.

Jonathan A. Barmak

Chapter 10. Fixed Points and the Lefschetz Number

In Chap. 9 we studied fixed point sets of group actions. Now we turn our attention to fixed point sets of continuous maps between finite spaces and their relationship with the fixed point sets of the associated simplicial maps. We analyze well-known results on the fixed point theory of finite posets from the perspective of finite spaces.

Jonathan A. Barmak

Chapter 11. The Andrews–Curtis Conjecture

The Poincaré conjecture is one of the most important problems in the history of Mathematics. The generalized versions of the conjecture for dimensions greater than 3 were proved between 1961 and 1982 by Smale, Stallings, Zeeman and Freedman. However, the original problem remained open for a century until Perelman finally proved it some years ago [66–68].

Jonathan A. Barmak


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