Qualifying Exam in Topology January 2006

... (a) If f is a closed map, then g is continuous. (b) If f is not a closed map, then g may fail to be continuous. 2. Let X = [0, 1]/( 41 , 34 ) be the quotient space of the unit interval, where the open interval ( 41 , 34 ) is identified to a single point. Show that: (a) X is connected. (b) X is compa ...

... (a) If f is a closed map, then g is continuous. (b) If f is not a closed map, then g may fail to be continuous. 2. Let X = [0, 1]/( 41 , 34 ) be the quotient space of the unit interval, where the open interval ( 41 , 34 ) is identified to a single point. Show that: (a) X is connected. (b) X is compa ...

Padic Homotopy Theory

... p-adic homotopy theory There should be p-adic homotopy theories for every prime p analogous to Sullivan’s Real homotopy theory. A norm on Q induces a unique topology on any finite dimensional vector V space over Q; hence V determines Vp a finite dimensional topological vector space over Qp. If V is ...

... p-adic homotopy theory There should be p-adic homotopy theories for every prime p analogous to Sullivan’s Real homotopy theory. A norm on Q induces a unique topology on any finite dimensional vector V space over Q; hence V determines Vp a finite dimensional topological vector space over Qp. If V is ...

HOMEWORK 7 Problem 1: Let X be an arbitrary nonempty set

... Remark: The discussion of Xtriv here is meant to illustrate a comment from lectures—if we drop the assumption of continuity from the definition of singular simplices, then the resulting “homology theory” is unable to distinguish any nonempty space from a singleton, in essence because the proof of ho ...

... Remark: The discussion of Xtriv here is meant to illustrate a comment from lectures—if we drop the assumption of continuity from the definition of singular simplices, then the resulting “homology theory” is unable to distinguish any nonempty space from a singleton, in essence because the proof of ho ...

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... g ◦ f ' idX , then f is a homotopy equivalence. This homotopy equivalence is sometimes called strong homotopy equivalence to distinguish it from weak homotopy equivalence. If there exist a homotopy equivalence between the topological spaces X and Y , we say that X and Y are homotopy equivalent, or t ...

... g ◦ f ' idX , then f is a homotopy equivalence. This homotopy equivalence is sometimes called strong homotopy equivalence to distinguish it from weak homotopy equivalence. If there exist a homotopy equivalence between the topological spaces X and Y , we say that X and Y are homotopy equivalent, or t ...

SPECTRA ACCORDING TO LURIE 0.1. This introduction is probably

... of Ek . (These arise, for example, by taking Ek to be the canonical k-plane bundle over the classifying space of some matrix group acting on Rk .) Such (pre)spectra are named after Thom for his applications to studying smooth cobordisms and characteristic classes. 2. Review of Loops and Suspensions ...

... of Ek . (These arise, for example, by taking Ek to be the canonical k-plane bundle over the classifying space of some matrix group acting on Rk .) Such (pre)spectra are named after Thom for his applications to studying smooth cobordisms and characteristic classes. 2. Review of Loops and Suspensions ...

2. Homeomorphisms and homotopy equivalent spaces. (14 October

... c) the sphere S n with identified diametrically opposed points (every pair of diametrically opposed points is identified); d) the disc Dn with identified diametrically opposed points of the boundary sphere S n−1 = ∂Dn . Problem 2. Prove that RP 1 ≈ S 1 . Problem 3. Prove that the space S 1 × I 1 is ...

... c) the sphere S n with identified diametrically opposed points (every pair of diametrically opposed points is identified); d) the disc Dn with identified diametrically opposed points of the boundary sphere S n−1 = ∂Dn . Problem 2. Prove that RP 1 ≈ S 1 . Problem 3. Prove that the space S 1 × I 1 is ...

MATH 6280 - CLASS 1 Contents 1. Introduction 1 1.1. Homotopy

... 1.2. CW-Complex and the Whitehead theorem. Definition 1.6. A CW–complex is a space built by gluing disks together along their boundary. If the largest kind of disk used to build X is Dn , then dim X = n. Example 1.7. The following admit the structure of CW–complexes: S n , CP n , RP n , differentiab ...

... 1.2. CW-Complex and the Whitehead theorem. Definition 1.6. A CW–complex is a space built by gluing disks together along their boundary. If the largest kind of disk used to build X is Dn , then dim X = n. Example 1.7. The following admit the structure of CW–complexes: S n , CP n , RP n , differentiab ...

Math 8301, Manifolds and Topology Homework 7

... 4. Suppose X is a topological space and φ : S n−1 → X is a continuous function, where n ≥ 1. Let Y = X ∪φ en be the space obtained from the disjoint union Dn ∪ X by identifying any point v ∈ S n−1 ⊂ Dn with its image φ(v) ∈ X. We say that Y is obtained from X by attaching an n-cell. Use the Seifert- ...

... 4. Suppose X is a topological space and φ : S n−1 → X is a continuous function, where n ≥ 1. Let Y = X ∪φ en be the space obtained from the disjoint union Dn ∪ X by identifying any point v ∈ S n−1 ⊂ Dn with its image φ(v) ∈ X. We say that Y is obtained from X by attaching an n-cell. Use the Seifert- ...

Lecture XI - Homotopies of maps. Deformation retracts.

... 1. Check that the map φ constructed in the proof of theorem 11.3 is continuous and is indeed a homotopy. Work out the proof of theorem 11.5. 2. Show that the boundary ∂M of the Möbius band M is not a deformation retract of M by taking a base point x0 on the boundary and computing explicitly the gro ...

... 1. Check that the map φ constructed in the proof of theorem 11.3 is continuous and is indeed a homotopy. Work out the proof of theorem 11.5. 2. Show that the boundary ∂M of the Möbius band M is not a deformation retract of M by taking a base point x0 on the boundary and computing explicitly the gro ...

k h b c b a q c p e a d r e m d f g n p r l m k g l q h n f

... homotopic to IdX and IdY , respectively. S 2 minus n points is homeomorphic to R2 minus n − 1 points, which deformation retracts onto a wedge of n − 1 copies of S 1 . The fundamental group of the Möbius band is Z, while that of the projective plane is Z/2. The two spaces are therefore not homotopy ...

... homotopic to IdX and IdY , respectively. S 2 minus n points is homeomorphic to R2 minus n − 1 points, which deformation retracts onto a wedge of n − 1 copies of S 1 . The fundamental group of the Möbius band is Z, while that of the projective plane is Z/2. The two spaces are therefore not homotopy ...

An introduction to equivariant homotopy theory Groups Consider

... An algebraic model, but not useful for calculations. Conjecture.(Greenlees) For any compact Lie group G there is an abelian category A(G) such that Q - G-spectra ' d. g. (A(G)) where A(G) has injective dimension equal to the rank of G. ...

... An algebraic model, but not useful for calculations. Conjecture.(Greenlees) For any compact Lie group G there is an abelian category A(G) such that Q - G-spectra ' d. g. (A(G)) where A(G) has injective dimension equal to the rank of G. ...

Midterm 1 solutions

... Let φ : X → Y be a continuous map. Another continuous map ψ : Y → X is called homotopy inverse for φ if ψ ◦ φ ' IdX and φ ◦ ψ ' IdY . If there exists a homotopy inverse for φ, then φ is called homotopy equivalence. In that case, we say that X is homotopy equivalent to Y (or X has the same homotopy t ...

... Let φ : X → Y be a continuous map. Another continuous map ψ : Y → X is called homotopy inverse for φ if ψ ◦ φ ' IdX and φ ◦ ψ ' IdY . If there exists a homotopy inverse for φ, then φ is called homotopy equivalence. In that case, we say that X is homotopy equivalent to Y (or X has the same homotopy t ...

Week 5: Operads and iterated loop spaces October 25, 2015

... The operadic point of view underscores a general philosophy in “homotopy coherent mathematics:" equalities between operations often must be replaced by homotopies. However, homotopies themselves are usually too unstructured a notion to be of much use. If, however, those homotopies are encoded as pa ...

... The operadic point of view underscores a general philosophy in “homotopy coherent mathematics:" equalities between operations often must be replaced by homotopies. However, homotopies themselves are usually too unstructured a notion to be of much use. If, however, those homotopies are encoded as pa ...

THE HIGHER HOMOTOPY GROUPS 1. Definitions Let I = [0,1] be

... The first part of the theorem is obvious while the second part is proved in much the same way as the analogous theorem was proved in the case of the fundamental group and is omitted. An important corollary of the above theorem is Corollary 2.4. If f : X → Y is a homeomorphism then f# : πn (X, p) → π ...

... The first part of the theorem is obvious while the second part is proved in much the same way as the analogous theorem was proved in the case of the fundamental group and is omitted. An important corollary of the above theorem is Corollary 2.4. If f : X → Y is a homeomorphism then f# : πn (X, p) → π ...

Problems for the exam

... 1. Let p ∈ CP2 and q ∈ RP3 . Is there a compact surface which is homotopy equivalent to CP2 \ {p} ? Is there a compact surface which is homotopy equivalent to RP3 \ {q} ? 2. Does the Borsuk-Ulam Theorem hold for the torus? In other words, is it true that for every continuous map f : S 1 × S 1 → R2 , ...

... 1. Let p ∈ CP2 and q ∈ RP3 . Is there a compact surface which is homotopy equivalent to CP2 \ {p} ? Is there a compact surface which is homotopy equivalent to RP3 \ {q} ? 2. Does the Borsuk-Ulam Theorem hold for the torus? In other words, is it true that for every continuous map f : S 1 × S 1 → R2 , ...

H-spaces II

... In this section we show that compact Lie groups other than SO(3)k × T l do admit special 1-tori. This implies that our theorem 1.4 is indeed stronger than theorem 1.1 of [8]. 5.1. Proposition. Let G be a compact connected Lie group which is not isomorphic to SO(3)k × T l . Then G has a subgroup isom ...

... In this section we show that compact Lie groups other than SO(3)k × T l do admit special 1-tori. This implies that our theorem 1.4 is indeed stronger than theorem 1.1 of [8]. 5.1. Proposition. Let G be a compact connected Lie group which is not isomorphic to SO(3)k × T l . Then G has a subgroup isom ...

Homotopy

... Two continuous functions from one topological space to another are called homotopic if one can be “continuously deformed” into the other, such a deformation being called a homotopy between the two functions. More precisely, we have the following definition. Definition 1.1. Let X, Y be topological sp ...

... Two continuous functions from one topological space to another are called homotopic if one can be “continuously deformed” into the other, such a deformation being called a homotopy between the two functions. More precisely, we have the following definition. Definition 1.1. Let X, Y be topological sp ...

Notes on wedges and joins

... union, and z0 is the point obtained by identification of x0 with y0 . The wedge sum is written as (X, x0 ) ∨ (Y, Y0 ), or simply as X ∨ Y . The latter notation is generally misleading, as we do not explicitly specify the basepoints x0 and y0 in X and Y . Choosing different basepoints x0 , y0 in X an ...

... union, and z0 is the point obtained by identification of x0 with y0 . The wedge sum is written as (X, x0 ) ∨ (Y, Y0 ), or simply as X ∨ Y . The latter notation is generally misleading, as we do not explicitly specify the basepoints x0 and y0 in X and Y . Choosing different basepoints x0 , y0 in X an ...

The Arithmetic Square (Lecture 32)

... Definition 2. Let f : X → Y be a map of topological spaces. We say that f is a rational homotopy equivalence if it induces an isomorphism on rational cohomology H∗ (Y ; Q) → H∗ (X; Q) (this is equivalent to the assertion that f induces an isomorphism on rational homology). We say that a space Z is r ...

... Definition 2. Let f : X → Y be a map of topological spaces. We say that f is a rational homotopy equivalence if it induces an isomorphism on rational cohomology H∗ (Y ; Q) → H∗ (X; Q) (this is equivalent to the assertion that f induces an isomorphism on rational homology). We say that a space Z is r ...

Let X be a path-connected space and suppose that every map f: S^1

... y in Y there exists a neighborhood Ny of b, so that p-1(Ny) is a disjoint union of sets Ai each of which is mapped homeomorphically onto Ny by map p. 3) If X is a contractible space then the fundamental group is trivial. 4) A topological space X is called simply connected if it is path-connected and ...

... y in Y there exists a neighborhood Ny of b, so that p-1(Ny) is a disjoint union of sets Ai each of which is mapped homeomorphically onto Ny by map p. 3) If X is a contractible space then the fundamental group is trivial. 4) A topological space X is called simply connected if it is path-connected and ...

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.