By Jonathan A. Barmak
This quantity bargains with the idea of finite topological areas and its
relationship with the homotopy and easy homotopy conception 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 crew 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 is usually instructed 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
B) Let K and L be finite simplicial complexes. Then, |K| we if X (K) ≈ X (L). 2 is one of the most useful tools to distinguish weak homotopy equivalences. Most of the times, we will apply this result to maps f : X → Y with Y ﬁnite, using the open cover given by the minimal basis of Y . 2 is closely related to the celebrated Quillen’s Theorem A, which gives a suﬃcient condition for a functor between two categories to be a homotopy equivalence at the level of classifying spaces . 2 also. It can be stated as follows.
If x is a point in an A-space X, the set Ux deﬁned as above is also open. The correspondence between ﬁnite spaces and ﬁnite preordered sets trivially extends to a correspondence between A-spaces and preordered sets. 6 shows that there is a weak homotopy equivalence |K(X)| → X. Conversely, given a simplicial complex K, the face poset X (K) is a locally ﬁnite space and 16 1 Preliminaries there is a weak homotopy equivalence |K| → X (K). Many of the results of this book can be stated in fact for A-spaces and general simplicial complexes.
We prove ﬁrst that Y is an open subspace of X. Suppose x = (h, i) ∈ Y . 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.