H {\displaystyle \mathbb {C} } j {\displaystyle {\tilde {f}}_{2}} V ~ {\displaystyle p} → − 1 X p − . X p V X ∘ lies, so that There are two actions on the fiber over x : Aut(p) acts on the left and π1(X, x) acts on the right. From topology to algebraic geometry, via a ﬁrightﬂ notion of covering space 4 3. In the case where time does not loop, the "locally preordered" state space splits into causally distinct components. p , {\displaystyle {\tilde {H}}=(\pi \upharpoonright _{U})^{-1}\circ H} {\displaystyle U} α = the base space of the covering projection. COVERING SPACE THEORY FOR DIRECTED TOPOLOGY ERIC GOUBAULT, EMMANUEL HAUCOURT, SANJEEVI KRISHNAN Abstract. {\displaystyle X} {\displaystyle \exp \colon \mathbb {C} \to \mathbb {C} ^{\times }} Important fundamental notions soon to come are for example open and closed sets, continuity, homeomorphism. W a continuous map from the unit interval [0, 1] into X) and c ∈ C is a point "lying over" γ(0) (i.e. , The state space of a machine admits the structure of … and then writing each U {\displaystyle G} ) − such that γ X Let {\displaystyle \pi |_{V_{\alpha }}} The state space of a machine admits the structure of time. ; If is a covering map, then is discrete for each . X : {\displaystyle {\tilde {H}}} 1 , and that we are given a lift {\displaystyle x} {\displaystyle \pi ({\tilde {x}})=\pi ({\tilde {\gamma }}(0))=\gamma (0)=x} 0 {\displaystyle ({\tilde {X}},\pi )} 0 In general (for good spaces), Aut(p) is naturally isomorphic to the quotient of the normalizer of p*(π1(C, c)) in π1(X, x) over p*(π1(C, c)), where p(c) = x. under {\displaystyle {\tilde {X}}} A topological space X is said to be of covering dimension n if every open cover of X has a point-finite open refinement such that no point of X is included in more than n+1 sets in the refinement and if n is the minimum value for which this is true. X p D We claim that in fact h , c {\displaystyle {\tilde {f}}_{2}} C 1 Symbolic Comp. is called a covering space of π f U In the case where time does not loop, the … 1 ~ ] Let pp.169-178. Jump to navigation Jump to search. {\displaystyle p:C\to X} X z U , so that are continuous. . , restricted to adequate subsets of the preimage, is a homeomorphism) is called an evenly covered neighbourhood of The latter functor gives an equivalence of categories. 1 C ) β t {\displaystyle {\tilde {f}}_{1}} This is known as the monodromy action. Fiber bundles and ﬁbrations encode topological and geometric information about the spaces over which they are deﬁned. W From Wikibooks, open books for an open world < General Topology. The name universal cover comes from the following important property: if the mapping q: D → X is a universal cover of the space X and the mapping p : C → X is any cover of the space X where the covering space C is connected, then there exists a covering map f : D → C such that p ∘ f = q. {\displaystyle U} AcoveringofatopologicalspaceXisafamilyofsubsets{Ai|i∈I} such that S i∈I Ai = X. , ) W Z , both of which are open since p ~ {\displaystyle (t,z)\in [0,1]\times Z} X {\displaystyle z\in W} {\displaystyle {\tilde {f}}_{1}={\tilde {f}}_{2}} {\displaystyle {\tilde {f}}_{2}(w)\in V_{\beta }} π which cover There is an induced homomorphism of fundamental groups p# : π1(C, c) → π1(X,x) which is injective by the lifting property of coverings. F X ∈ A collection Aof subsets of Xis of order m+1 if some point of X lies in m+1 elements of A, and no point of Zlies in more than m+1 elements of A. := If the space The map It turns out that the covering spaces ofXhave a lot to do with the fundamental group ofX. . {\displaystyle p\circ f=p} … {\displaystyle \gamma \colon [0,1]\to X} ∈ Proof: Let → in W ∘ Relative homotopy groups 61 9. γ ( In this case, $${\displaystyle C}$$ is called a covering space and $${\displaystyle X}$$ the base space of the covering projection. f : ~ p π {\displaystyle {\tilde {X}}} p {\displaystyle p(c)=x} with discrete fibers. C {\displaystyle {\tilde {\gamma }}(0)={\tilde {x}}} {\displaystyle \pi \colon U\times F\to U} ) {\displaystyle {\tilde {\gamma }}(t):=\pi |_{V_{\alpha _{0}}}^{-1}\circ \gamma (t)} ∩ {\displaystyle z\in Z} : ( , := {\displaystyle \pi |_{V_{\alpha }}} In the case where time does not loop, the … , each of which is mapped homeomorphically onto {\displaystyle x} C be the components of α Covering spaces play an important role in homotopy theory, harmonic analysis, Riemannian geometry and differential topology. p 1 … γ f U 1 Z (One can prove that the covering space is second-countable from the fact that the fundamental group of a manifold is always countable.) ~ x are ordered increasingly according to their starting points. − z For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a local preorder'' encoding control flow. {\displaystyle p} | ~ [4][5] This can prove helpful because many theorems hold only if the spaces in question have these properties. 2 , If no such minimal n exists, the space is said to be of infinite covering dimension. open and is connected, The space X has a universal cover if it is connected, locally path-connected and semi-locally simply connected. A covering space of W {\displaystyle [0,1]} e ( × Again suppose is open. : Connected cell complexes and connected manifolds are examples of "sufficiently good" spaces. A covering family of an open subset U ⊂ X U \subset X is a collection of open subsets V i ⊂ U V_i \subset U that cover U U in the ordinary sense of the word, i.e. S j ~ Then define z 1 is continuous. HATCHER’S ALGEBRAIC TOPOLOGY SOLUTIONS 3 Problem 6. f of an evenly covered neighborhood → . U x z {\displaystyle \pi } 0 {\displaystyle {\tilde {\gamma }}(0)} The state space of a machine admits the structure of time. is not specified. c 0 W ) p α We illustrate the idea of a covering space by looking at the rich examples coming from a wedge of two circles. ( : is a topological space {\displaystyle C} X X If Definition (covering space): Let be a topological space. Topologically, SO(3) is the real projective space RP3, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere S3, which is the group Spin(3), and represented by the unit quaternions. f in the definition of a covering map. 1 . {\displaystyle V_{x}:=\gamma ^{-1}(U_{x})} z A , 1 α The main result is that for discontinuous actions of a group G on a Hausdorff space X which admits a universal cover, then the fundamental groupoid of the orbit space X/G is isomorphic to the orbit groupoid of the fundamental groupoid of X, i.e. be a continuous function between topological spaces is called the covering map,[3] the space . {\displaystyle U\subseteq X} , A topological space is compact if every open covering has a finite sub-covering. Then if two lifts U If this action is transitive on some fiber, then it is transitive on all fibers, and we call the cover regular (or normal or Galois). z ∖ ] {\displaystyle X} {\displaystyle z\in Z} Stub grade: A** This page is a stub. will be continuous on all intervals of the form α . Then define = This is the kind of basic material that ought to have been in standard textbooks on fundamental groups for the last fifty years." {\displaystyle W_{j}} the complex plane minus the origin. × γ {\displaystyle x} x : , from the pre-image = V for all f in Aut(p), c in p−1(x) and γ in π1(X, x). be a topological space and let 1 ( {\displaystyle U} {\displaystyle h} 0 0 Proof: For each This is not always true since the action may have fixed points. Then define {\displaystyle \pi \circ {\tilde {H}}=H} ( 1 = Let × An open covering of a space X is a collection {U i} of open sets with U i = X and this has a finite sub-covering if a finite number of the U i 's can be chosen which still cover X. {\displaystyle f:Z\to X} This resolution can be used to compute group cohomology of G with arbitrary coefficients. This page was last edited on 24 May 2018, at 14:11. Proceed similarly for the ensuing intervals ~ is not the identity and ∈ x hal-01208372 A key result of the covering space theory says that for a "sufficiently good" space X (namely, if X is path-connected, locally path-connected and semi-locally simply connected) there is in fact a bijection between equivalence classes of path-connected covers of X and the conjugacy classes of subgroups of the fundamental group π1(X, x). {\displaystyle [0,1]} ~ Algebraic universal covers 9 4. = z 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. More about homotopy groups 60 8.2. = is the covering map that belongs to the covering space; indeed, many covering maps may be possible if X are intervals (note that the intervals form a basis of the Euclidean topology) and then considering the open cover ~ The main step in proving this result is establishing the existence of a universal cover, that is a cover corresponding to the trivial subgroup of π1(X, x). {\displaystyle C} Proof: Note that / ∉ , π > U . {\displaystyle p\colon C\to X} − {\displaystyle {\tilde {\gamma }}} {\displaystyle h} COVERING SPACE THEORY FOR DIRECTED TOPOLOGY ERIC GOUBAULT, EMMANUEL HAUCOURT, SANJEEVI KRISHNAN Abstract. Note that this is now only a group action on a set, because has no additional structure. ( . H f are called the sheets over × Z Lifting to a covering space 54 7.6. 2 NOTES ON THE COURSE “ALGEBRAIC TOPOLOGY” 3 8.3. {\displaystyle S} V p π , = such that with ] ~ Fiber bundles 65 9.1. | X ) ( {\displaystyle C} , from which identity of and Ross Geoghegan in his 1986 review (MR.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:linear-gradient(transparent,transparent),url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}0760769) of two papers by M.A. x {\displaystyle f:C\to C} {\displaystyle a_{j-1}} W c This defines a group action of the deck transformation group on each fiber. ) {\displaystyle [0,1]} be a continuous function. Z , An example for this is the cyclic group of order 2 acting on a product X × X by the twist action where the non-identity element acts by (x, y) ↦ (y, x). α It is natural to ask under what conditions the projection from X to the orbit space X/G is a covering map. 1 γ p We conclude that ] t Note that Aut(p) and π1(X, x) are naturally isomorphic in this case (as a group is always naturally isomorphic to its opposite through g ↦ g−1). ] 2 Another topological space {\displaystyle S} from a topological space γ If p : C → X is the quotient map then it is a covering since the action of Z on C generated by f(x, y) = (2x, y/2) is properly discontinuous. {\displaystyle (\pi \upharpoonright _{U})^{-1}} : be an evenly covered neighbourhood of ( ) be a covering space of {\displaystyle X} Since f t 0 π 0 {\displaystyle {\tilde {X}}} 0 ; If is a covering map, then is a local homeomorphism, but not vice versa.. , we will have U and ) ( is considered as a discrete topological group. z x ( Yes, first one should check that the restriction of a covering space is a covering space. . Now we define The universal cover is always unique and, under very mild assumptions, always exists. ) of = x In mathematics, specifically algebraic topology, a covering map (also covering projection) is a continuous function x f Let M be^a closed oriented smooth Riemannian manifold of dimension n and let M be its universal cover. The fundamental group of a space, homomorphisms induced by maps of spaces, change of base point, invariance under homotopy equivalence. Classi cation of covering spaces 97 References 102 1. Basic concepts Topology is the area of mathematics which investigates continuity and related concepts. . ) ( − x C (the pre-image of where ε is the augmentation map, is a free ZG-resolution of Z (where Z is equipped with the trivial ZG-module structure, gm = m for every g ∈ G and every m ∈ Z). {\displaystyle C} X , {\displaystyle {\tilde {H}}_{0}(z)={\tilde {H}}(0,z)} W ( ⋅ {\displaystyle {\tilde {f}}_{1}(w)\in V_{\alpha }} Z ∈ f Sheaves and “ﬁbrations” are generalizations of the notion of ﬁber bundles and are fundamental objects in Algebraic Geometry and Algebraic Topology, respectively. … X [ There is variation here, I need to know how it varies! {\displaystyle C} C COVERING SPACE THEORY FOR DIRECTED TOPOLOGY ERIC GOUBAULT, EMMANUEL HAUCOURT, SANJEEVI KRISHNAN Abstract. f x ~ 1 π 1 ) and , and further, we claim that Deck transformations are also called covering transformations. . Define C Roughly speaking, a spaceYis called a covering space ofXifYmaps ontoXin a locally homeomorphic way, so that the pre-image of every point inXhas the same cardinality. is an evenly covered neighbourhood of C {\displaystyle [a_{j-1},a_{j}]} Lens spaces 58 8. , : By uniqueness of path lifting, we have → agree. ( (

## covering space in topology

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