By Jonathan A. Barmak
This quantity offers with the speculation of finite topological areas and its
relationship with the homotopy and straightforward homotopy idea 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 constructed 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 really 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 resources for Algebraic Topology of Finite Topological Spaces and Applications
These spaces play a fundamental role in the theory of ﬁnite spaces. 10 we will prove that if X and Y are ﬁnite T0 -spaces, there is a weak homotopy equivalence |K(X)| ∨ |K(Y )| → X ∨ Y . 8 A Finite Analogue of the Mapping Cylinder The mapping cylinder of a map f : X → Y between topological spaces is the space Zf obtained from (X × I) Y by identifying each point (x, 1) ∈ X × I with f (x) ∈ Y . Both X and Y are subspaces of Zf . We denote by j : Y → Zf and i : X → Zf the canonical inclusions where i is deﬁned by i(x) = (x, 0).
Him for some 1 ≤ i1 , i2 . . im ≤ r. On the other hand, (khi1 hi2 . . his , −1) and (khi1 hi2 . . his+1 , −1) are connected via (khi1 hi2 . . his , −1) < (khi1 hi2 . . his , r) > (khi1 hi2 . . his+1 , −1). This implies that (k, −1) and (h, −1) are in the same connected component. e. f = φ(g). Therefore φ is an epimorphism, and then G Aut(X). 7 Joins, Products, Quotients and Wedges 29 If the generators h1 , h2 , . . , hr are non-trivial, the open sets U(g,r) are as in Fig. 1. In that case it is not hard to prove that the ﬁnite space X constructed above is weak homotopy equivalent to a wedge of n(r − 1) + 1 circles, or in other words, that the order complex of X is homotopy equivalent to a wedge of n(r − 1) + 1 circles.
2. For example, for any n ≥ 1, the n-dimensional sphere S n does not have the homotopy type of any ﬁnite space. However, S n does have, as any ﬁnite polyhedron, the same weak homotopy type as some ﬁnite space. 4 Loops in the Hasse Diagram and the Fundamental Group In this section we give a full description of the fundamental group of a ﬁnite T0 -space in terms of its Hasse diagram. This characterization is induced from the well known description of the fundamental group of a simplicial complex.