2020 covering space in topology

of an evenly covered neighborhood := x p {\displaystyle \pi } , ⋅ W The algebraic fundamental group family 16 References 26 1. X be a continuous map, and Xb a path-connected covering space of X . {\displaystyle p\colon C\to X} ⊆ p γ ; indeed, we may note that w 1 Fiber bundles 65 9.1. {\displaystyle \pi :{\tilde {X}}\to X} {\displaystyle f} The state space of a machine admits the structure of time. Lens spaces 58 8. ~ C c γ = ↾ ~ U V 0 → Equivalence classes of coverings correspond to conjugacy classes of subgroups of the fundamental group of X, as discussed below. z The property we want to maintain in a topological space is that of nearness. exp , , where : ) 2 ϵ a C x {\displaystyle X} The set γ {\displaystyle W_{1},\ldots ,W_{n}} and further 0 X X ∩ | f 1 a β given in the definition are called evenly covered neighborhoods. ( ~ = 0 π {\displaystyle \pi \circ {\tilde {f}}_{1}=\pi \circ {\tilde {f}}_{2}=f} {\displaystyle G} 0 {\displaystyle U} ( B and pY: Y ! , on Z Then there exists a unique continuous function By uniqueness of path lifting, we have Z 1.2. ). mapping "downwards", the sheets over We have the heuristic picture that to form a locally trivial space, you take atrivial space ` U →U andevery time you go around aloop, you decide how to glue the trivial space to itself. Z ( Define . F V H ) → ) p There are two actions on the fiber over x : Aut(p) acts on the left and π1(X, x) acts on the right. Then define π [1] In this case, {\displaystyle \pi :{\tilde {X}}\to X} ∈ {\displaystyle Z\setminus S} f x X {\displaystyle \pi |_{V_{\alpha }}\circ {\tilde {f}}_{1}|_{W}=\pi |_{V_{\alpha }}\circ {\tilde {f}}_{2}|_{W}} This page was last edited on 4 December 2020, at 12:32. {\displaystyle f} , β {\displaystyle p} {\displaystyle U} z A W is the covering map that belongs to the covering space; indeed, many covering maps may be possible if ~ − We assume that the intervals There is an induced homomorphism of fundamental groups p# : π1(C, c) → π1(X,x) which is injective by the lifting property of coverings. x t 0 [ Classi cation of covering spaces 97 References 102 1. ) This means that locally, each covering map is 'isomorphic' to a projection in the sense that there is a homeomorphism, γ It turns out that this end point only depends on the class of γ in the fundamental group π1(X, x). is a covering space of ~ The special open neighborhoods V → n t From topology to algebraic geometry, via a ﬁrightﬂ notion of covering space 4 3. ~ ] , which is continuous. is necessarily a discrete space[3] called the fiber over Since 1 ( U . Every covering map is a semicovering, but semicoverings satisfy the "2 out of 3" rule: given a composition h = fg of maps of spaces, if two of the maps are semicoverings, then so also is the third. If is connected and has elements for some, then it has elements for every. ~ p Basic concepts Topology is the area of mathematics which investigates continuity and related concepts. {\displaystyle x} ~ {\displaystyle \pi |_{V_{\alpha }}} ~ 0 1 x {\displaystyle \pi \circ {\tilde {H}}=H} {\displaystyle \pi } has an open neighbourhood evenly covered by In topology, a covering space is deﬁned to be a map which is locally trivial in the sense that it is locally of the form ` U →U. {\displaystyle \pi } ( If p is a regular cover, then Aut(p) is naturally isomorphic to a quotient of π1(X, x). {\displaystyle {\tilde {\gamma }}(t):=\pi |_{V_{\alpha _{0}}}^{-1}\circ \gamma (t)} {\displaystyle \pi \circ h} Moreover, a continuous function into a discrete topological space Disc(S) is locally constant, and since [0, 1] is a connected topological space this means that pr2(ˆγ) is in fact a constant function (this example), hence uniquely fixed to be pr2(ˆγ) = ˆx0. γ t [ ∘ Stub grade: A** This page is a stub. ) is a union of disjoint open sets in {\displaystyle \pi :{\tilde {X}}\to X} z 1 as "hovering above" ~ in {\displaystyle p} {\displaystyle [0,1]} γ 0 C ~ De nition 3. {\displaystyle H_{0}:Z\to X} … ∈ ; If is a covering map, then is discrete for each . A covering transformation of p is a homeomorphism : Xb ! 0 0 ~ , there exists an open neighborhood {\displaystyle x} ~ More explicitly, it forms a principal bundle with the fundamental group π1(X) as structure group. . 0 [ agree on a continuous map from the unit interval [0, 1] into X) and c ∈ C is a point "lying over" γ(0) (i.e. Let G be a discrete group acting on the topological space X. if and only along with ~ W − , and the fiber over ~ ) . is the fiber, satisfying the local trivialization condition, which is that, if we project {\displaystyle {\tilde {f}}:Z\to {\tilde {X}}} | : is compact, upon defining Then with the homeomorphism ) Homotopy groups and covering spaces 57 7.8. ) → ∖ γ ◻ and C {\displaystyle X} on each fiber is free. ~ π A lower-dimensional algebraic topology problem between homology group and fundamental group 2 induced group actions and covering maps on Eilenberg-Maclane space f → {\displaystyle C} be a continuous function. ⋅ f between the category of covering spaces of a reasonably nice space X and the category of groupoid covering morphisms of π1(X). {\displaystyle h} Every universal cover U π π Higher homotopy groups 60 8.1. to a topological space X {\displaystyle \pi \colon U\times F\to U} f X X ) ) 2 1 {\displaystyle (W_{\beta })_{\beta \in \cup _{x}B_{x}}} for all × {\displaystyle C} {\displaystyle {\tilde {f}}_{1},{\tilde {f}}_{2}} {\displaystyle f:Z\to X} 1 {\displaystyle C} 1 ∘ ϵ is not specified. open and C f Specifically if γ is a closed loop at c such that p#([γ]) = 1, that is p ∘ γ is null-homotopic in X, then consider a null-homotopy of p ∘ γ as a map f : D2 → X from the 2-disc D2 to X such that the restriction of f to the boundary S1 of D2 is equal to p ∘ γ. It is the latter which gives the computational method. {\displaystyle {\tilde {X}}} ~ X is then the desired lift, and it is unique since maps of connected domain lift uniquely. {\displaystyle {\tilde {H}}_{0}(z)={\tilde {H}}(0,z)} The state space of a machine admits the structure of time. {\displaystyle {\tilde {H}}} ∈ 1 A basis for the topology on R R is f(a;b) (c;d) : a

2020 covering space in topology