THE FUNDAMENTAL GROUP, COVERING SPACES AND
... We can now look at contractible spaces as being a special case of a more general class of spaces. Definition 5 (Simply Connected Space). A topological space (X, A) is said to be a simply connected space if it is path connected and the fundamental group of X is trivial. A simply connected space is th ...
... We can now look at contractible spaces as being a special case of a more general class of spaces. Definition 5 (Simply Connected Space). A topological space (X, A) is said to be a simply connected space if it is path connected and the fundamental group of X is trivial. A simply connected space is th ...
AAG, LECTURE 13 If 0 → F 1 → F2 → F3 → 0 is a short exact
... but it is not necessarily the case that F2 (X) → F3 (X) is surjective. (The surjectivity of F2 → F3 implies something weaker: that for any f ∈ F3 (X) and x ∈ X there is an open x ∈ U ⊂ X and a g ∈ F2 (U ) so that g 7→ f|U . These g’s are not unique, since we can change them by sections of F1 , so on ...
... but it is not necessarily the case that F2 (X) → F3 (X) is surjective. (The surjectivity of F2 → F3 implies something weaker: that for any f ∈ F3 (X) and x ∈ X there is an open x ∈ U ⊂ X and a g ∈ F2 (U ) so that g 7→ f|U . These g’s are not unique, since we can change them by sections of F1 , so on ...
CLASSIFYING SPACES OF MONOIDS – APPLICATIONS IN
... was proved in [A-M-S], that D2 is the well-known Dunce Hat, and that each D2n , for n > 1, is a contractible, not collapsible polyhedron. Consequently, D2n , for n > 2, was refereed to as a Higher-dimensional Dunce Hat. Our basic observation, based on the original definition of Dn given in [A-M-S] ( ...
... was proved in [A-M-S], that D2 is the well-known Dunce Hat, and that each D2n , for n > 1, is a contractible, not collapsible polyhedron. Consequently, D2n , for n > 2, was refereed to as a Higher-dimensional Dunce Hat. Our basic observation, based on the original definition of Dn given in [A-M-S] ( ...
Homotopy type of symplectomorphism groups of × S Geometry & Topology
... homotopy equivalent to its subgroup of standard isometries SO(3) × SO(3). He also showed that this would no longer hold when one sphere is larger than the other, and in [9] McDuff constructed explicitly an element of infinite order in H1 (Gλ ), λ > 0. The main tool in their proofs is to look at the ...
... homotopy equivalent to its subgroup of standard isometries SO(3) × SO(3). He also showed that this would no longer hold when one sphere is larger than the other, and in [9] McDuff constructed explicitly an element of infinite order in H1 (Gλ ), λ > 0. The main tool in their proofs is to look at the ...
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... subgroup that consists of all the degenerate elements, which are finite linear combinations of elements of the form si (y). Note that commutation relations between di and si ensure that the map d : An (K) → An−1 (K) descends to the quotient, and we get a map d : Cn (K) → Cn−1 (K). This is equivalent ...
... subgroup that consists of all the degenerate elements, which are finite linear combinations of elements of the form si (y). Note that commutation relations between di and si ensure that the map d : An (K) → An−1 (K) descends to the quotient, and we get a map d : Cn (K) → Cn−1 (K). This is equivalent ...
Stable ∞-Categories (Lecture 3)
... It turns out that for any topological category C, the homotopy coherent nerve Nt (C) is an ∞-category. Moreover, there is a sort of converse: every ∞-category is equivalent to Nt (C) for some topological category C, and the topological category C is essentially unique (up to a suitable notion of we ...
... It turns out that for any topological category C, the homotopy coherent nerve Nt (C) is an ∞-category. Moreover, there is a sort of converse: every ∞-category is equivalent to Nt (C) for some topological category C, and the topological category C is essentially unique (up to a suitable notion of we ...
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... is the content of Section 3. Constructing the homology groups in this manner at first seems counterproductive, as it does not capitalize on the advantage of the homology groups, that they can be more easily computed, but if anything nullifies this advantage by giving them as homotopy groups of certa ...
... is the content of Section 3. Constructing the homology groups in this manner at first seems counterproductive, as it does not capitalize on the advantage of the homology groups, that they can be more easily computed, but if anything nullifies this advantage by giving them as homotopy groups of certa ...
L19 Abstract homotopy theory
... should really be concerned about whether the morphisms just defined are really a set, but this turns out not to be important.) The homotopy category also has an equivalent description that looks more like taking homotopy classes of maps. Namely, suppose we wish to define the homotopy equivalence rel ...
... should really be concerned about whether the morphisms just defined are really a set, but this turns out not to be important.) The homotopy category also has an equivalent description that looks more like taking homotopy classes of maps. Namely, suppose we wish to define the homotopy equivalence rel ...
The homotopy category is a homotopy category. Arne Str¢m
... would then be the ordinary homotopy category of topological spaces, i.e. the objects would be all topological spaces and the morphisms would be all homotopy classes of continuous maps. In the first section of this paper we prove that this is indeed feasible, and in the last section we consider the c ...
... would then be the ordinary homotopy category of topological spaces, i.e. the objects would be all topological spaces and the morphisms would be all homotopy classes of continuous maps. In the first section of this paper we prove that this is indeed feasible, and in the last section we consider the c ...
ALGEBRAIC TOPOLOGY Contents 1. Preliminaries 1 2. The
... As we hinted in our intuitive explanation about the fundamental group of the circle, a space X and its covering space are related by what are called lifts. Lifts give us the correspondence between paths in X and paths in X̃. Definition 4.3. Let X, Y be spaces and p : X̃ → X be a covering space of X. ...
... As we hinted in our intuitive explanation about the fundamental group of the circle, a space X and its covering space are related by what are called lifts. Lifts give us the correspondence between paths in X and paths in X̃. Definition 4.3. Let X, Y be spaces and p : X̃ → X be a covering space of X. ...
Algebraic Topology Introduction
... Y to X, so take h = f −1 as in the definition of homotopy equivalence. Then, not only are f ◦ f −1 and f −1 ◦ f homotopic to the respective identity maps, they are on the nose equal. The converse is certainly not true. That is, if X and Y are homotopy equivalent spaces, they are not necessarily home ...
... Y to X, so take h = f −1 as in the definition of homotopy equivalence. Then, not only are f ◦ f −1 and f −1 ◦ f homotopic to the respective identity maps, they are on the nose equal. The converse is certainly not true. That is, if X and Y are homotopy equivalent spaces, they are not necessarily home ...
THE WEAK HOMOTOPY EQUIVALENCE OF Sn AND A SPACE
... and downbeat points to obtain XT0 ,min which is also homotopically equivalent to X. We write this down as a theorem just to record our achievements. Theorem 2.7. Given a finite space, X, there exists a finite minimal T0 -space, XT0 ,min , which is homotopically equivalent to X. In fact, this minimal ...
... and downbeat points to obtain XT0 ,min which is also homotopically equivalent to X. We write this down as a theorem just to record our achievements. Theorem 2.7. Given a finite space, X, there exists a finite minimal T0 -space, XT0 ,min , which is homotopically equivalent to X. In fact, this minimal ...
23 Introduction to homotopy theory
... Example 23.3. One can define a model for S starting from simplicial sets sSet := Set . Recall that op is the category whose objects are the posets [n] = {0, . . . , n} and whose ...
... Example 23.3. One can define a model for S starting from simplicial sets sSet := Set . Recall that op is the category whose objects are the posets [n] = {0, . . . , n} and whose ...
Some results in quasitopological homotopy groups
... COROLLARY 2.4. Let X be a topological space such that Ωn−1 (X, x) is small generated. Then πnqtop (X, x) is an indiscrete topological group. Proof. Since Ωn−1 (X, x) is a small generated space, then π1qtop (Ωn−1 (X, x), ex ) is an indisqtop crete topological group, by [15, Remark 2.11]. Therefore π ...
... COROLLARY 2.4. Let X be a topological space such that Ωn−1 (X, x) is small generated. Then πnqtop (X, x) is an indiscrete topological group. Proof. Since Ωn−1 (X, x) is a small generated space, then π1qtop (Ωn−1 (X, x), ex ) is an indisqtop crete topological group, by [15, Remark 2.11]. Therefore π ...
Orbit Projections as Fibrations
... in each Gx-space V (resp. V + ) are compact manifolds and therefore CW complexes. It follows that the arguments in 2.2, 2.3, and 2.4 are applicable and we obtain Theorem. Let M be a proper locally smooth G-space (with boundary). The orbit projection π : M → M/G is a Hurewicz (Serre) (G-)fibration if ...
... in each Gx-space V (resp. V + ) are compact manifolds and therefore CW complexes. It follows that the arguments in 2.2, 2.3, and 2.4 are applicable and we obtain Theorem. Let M be a proper locally smooth G-space (with boundary). The orbit projection π : M → M/G is a Hurewicz (Serre) (G-)fibration if ...
SAM III General Topology
... To a topological space X = (X , τ ) one associates a category Π1 X defined as follows: objects of the category are points in the topological space, while a morphism from x to y is a homotopy class [f ] of a path f from x to y ; composition of morphisms is defined by the formula [g ] ◦ [f ] = [g ◦ f ...
... To a topological space X = (X , τ ) one associates a category Π1 X defined as follows: objects of the category are points in the topological space, while a morphism from x to y is a homotopy class [f ] of a path f from x to y ; composition of morphisms is defined by the formula [g ] ◦ [f ] = [g ◦ f ...
Elements of Homotopy Fall 2008 Prof. Kathryn Hess Series 13 Let B
... (b) If B is path connected and E is nonempty, then p is a surjective map. Exercise 4. Let p : E−→B be a fibration, b0 ∈ B, and F := p−1 (b0 ) ⊆ E the fiber over b0 . Assume F is nonempty. Denote by i : F −→E the inclusion map. Prove the following: (a) If B is path connected, then the induced map π0 ...
... (b) If B is path connected and E is nonempty, then p is a surjective map. Exercise 4. Let p : E−→B be a fibration, b0 ∈ B, and F := p−1 (b0 ) ⊆ E the fiber over b0 . Assume F is nonempty. Denote by i : F −→E the inclusion map. Prove the following: (a) If B is path connected, then the induced map π0 ...
Pointed spaces - home.uni
... Example 2.2. U : Gp → Set the underlying set functor, which associates to a group its underlying set. Remark 2.3. The functor U forgets the group structure of G. For this reason, a functors of that type is often called a forgetful functor. Example 2.4. F : Set → Gp the free group functor. Remark 2.5 ...
... Example 2.2. U : Gp → Set the underlying set functor, which associates to a group its underlying set. Remark 2.3. The functor U forgets the group structure of G. For this reason, a functors of that type is often called a forgetful functor. Example 2.4. F : Set → Gp the free group functor. Remark 2.5 ...
Lecture 4(30.01.09) Universal Bundles
... CP3 , and their union is homotopy equivalent to ΣP3 . This lemma implies that Ek is homotopy equivalent to a k-fold suspension over some space and, thus, is k − 1-connected. Theorem 2 For any countable, connected simplicial complex X in the weak topology (closed sets are those with closed intersecti ...
... CP3 , and their union is homotopy equivalent to ΣP3 . This lemma implies that Ek is homotopy equivalent to a k-fold suspension over some space and, thus, is k − 1-connected. Theorem 2 For any countable, connected simplicial complex X in the weak topology (closed sets are those with closed intersecti ...
Homotopy groups of spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.The n-dimensional unit sphere — called the n-sphere for brevity, and denoted as Sn — generalizes the familiar circle (S1) and the ordinary sphere (S2). The n-sphere may be defined geometrically as the set of points in a Euclidean space of dimension n + 1 located at a unit distance from the origin. The i-th homotopy group πi(Sn) summarizes the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn. This summary does not distinguish between two mappings if one can be continuously deformed to the other; thus, only equivalence classes of mappings are summarized. An ""addition"" operation defined on these equivalence classes makes the set of equivalence classes into an abelian group.The problem of determining πi(Sn) falls into three regimes, depending on whether i is less than, equal to, or greater than n. For 0 < i < n, any mapping from Si to Sn is homotopic (i.e., continuously deformable) to a constant mapping, i.e., a mapping that maps all of Si to a single point of Sn. When i = n, every map from Sn to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies πn(Sn) with the group of integers under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular mapping. However, the most interesting and surprising results occur when i > n. The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S3 around the usual sphere S2 in a non-trivial fashion, and so is not equivalent to a one-point mapping.The question of computing the homotopy group πn+k(Sn) for positive k turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups πn+k(Sn) are independent of n for n ≥ k + 2. These are called the stable homotopy groups of spheres and have been computed for values of k up to 64. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups (for n < k + 2) are more erratic; nevertheless, they have been tabulated for k < 20. Most modern computations use spectral sequences, a technique first applied to homotopy groups of spheres by Jean-Pierre Serre. Several important patterns have been established, yet much remains unknown and unexplained.