Finite Spaces and Simplicial Complexes
ثبت نشده
چکیده
Finite simplicial complexes provide a general class of spaces that is sufficient for most purposes of basic algebraic topology. There are more general classes of spaces, in particular the finite CW complexes, that are more central to the modern development of the subject, but they give exactly the same collection of homotopy types. The relevant background on simplicial complexes will be recalled as we go along and can be found in most textbooks in algebraic topology (but not in my own book [7]). We write |K| for the geometric realization of K. We recall the definition of the homotopy groups πn(X,x) of a space X at x ∈ X. When n = 0, this is just the set of path components of X, with the component of x taken as a basepoint (and there is no group structure). When n = 1 it is the fundamental group of X at the point x. For all n ≥ 0, it can be described most simply by considering the standard sphere S with a chosen basepoint ∗. One considers all maps α : S −→ X such that f(∗) = x. One says that two such maps α and β are based homotopic if there is a based homotopy h : α ' β. Here a homotopy h is based if h(∗, t) = x for all t ∈ I. If n = 1, the map α is a loop at x, and we can compose loops to obtain a product which makes π1(X,x) a group. The homotopy class of the constant loop at x gives the identity element, and the loop α−1(t) = α(1− t) represents the inverse of the homotopy class of α. There is a similar product on the higher homotopy groups, but, in contrast to the fundamental group, the higher homotopy groups are Abelian.
منابع مشابه
Finite Spaces and Applications to the Euler Characteristic
The aim of this paper is to introduce finite spaces and their simplicial complexes. Next, we give an application to combinatorics in the form of a relation between the Euler characteristic and the Möbius function. We begin by giving an overview of finite topological spaces. We introduce beat points and weak homotopy equivalences. Then, we show that finite spaces are weak homotopy equivalent to ...
متن کاملSimple Homotopy Types and Finite Spaces
We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse $X\searrow Y$ of finite spaces induces a simplicial collapse $\k(X)\searrow \k(Y)$ of their associated simplicial c...
متن کاملFinite Spaces and Quillen’s Conjecture
We introduce finite spaces and explain their relations to sets with reflexive and transitive relations and to simplicial complexes. We introduce Quillen’s conjecture: If Ap(G) is weakly contractible then G has a nontrivial normal p-subgroup. We use Quillen’s method to show this is true for solvable groups.
متن کاملVertex Decomposable Simplicial Complexes Associated to Path Graphs
Introduction Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when . Later Bjorner and Wachs extended this concept to non-pure ...
متن کاملCohen-Macaulay-ness in codimension for simplicial complexes and expansion functor
In this paper we show that expansion of a Buchsbaum simplicial complex is $CM_t$, for an optimal integer $tgeq 1$. Also, by imposing extra assumptions on a $CM_t$ simplicial complex, we provethat it can be obtained from a Buchsbaum complex.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010