IV.2 Homology
... point then A is well defined and has degree 0 because it extends a map from the ball to the sphere. We now construct H : Sd × [0, 1] → Sd defined by H(x, t) = (x − tf (x))/kx − tf (x)k. For t = 1 we have x 6= f (x) because there is no fixed point and for t < 1 we have x 6= tf (x) because kxk = 1 > k ...
... point then A is well defined and has degree 0 because it extends a map from the ball to the sphere. We now construct H : Sd × [0, 1] → Sd defined by H(x, t) = (x − tf (x))/kx − tf (x)k. For t = 1 we have x 6= f (x) because there is no fixed point and for t < 1 we have x 6= tf (x) because kxk = 1 > k ...
Embeddings vs. Homeomorphisms (Lecture 13)
... Rn to itself: Theorem 1 is equivalent to the assertion that the inclusion between these topological spaces is a weak homotopy equivalence. Remark 3. We can also define simplicial sets which parametrize PL embeddings and PL homeomorphisms from Rn to itself. Theorem 1 continues to hold in this case, u ...
... Rn to itself: Theorem 1 is equivalent to the assertion that the inclusion between these topological spaces is a weak homotopy equivalence. Remark 3. We can also define simplicial sets which parametrize PL embeddings and PL homeomorphisms from Rn to itself. Theorem 1 continues to hold in this case, u ...
Homotopy Theory
... This just follows from the theorem and the definitions. Definition 7.11. Map0 (Y, Z) is the space of all pointed maps Y → Z with the compact-open topology. This is a subspace of Map(Y, Z). The base point of Map0 (Y, Z) is the mapping ∗ which sends all of Y to the base point of Z. 7.1.2. fancy versio ...
... This just follows from the theorem and the definitions. Definition 7.11. Map0 (Y, Z) is the space of all pointed maps Y → Z with the compact-open topology. This is a subspace of Map(Y, Z). The base point of Map0 (Y, Z) is the mapping ∗ which sends all of Y to the base point of Z. 7.1.2. fancy versio ...
Homotopy Theory of Finite Topological Spaces
... intuition, namely that a finite space is endowed with the discrete topology (as it would be, for example, if it were a finite subset of Euclidean space with the subspace topology). Upon a moment’s further reflection, however, it is apparent that this is by no means necessarily the case, and there co ...
... intuition, namely that a finite space is endowed with the discrete topology (as it would be, for example, if it were a finite subset of Euclidean space with the subspace topology). Upon a moment’s further reflection, however, it is apparent that this is by no means necessarily the case, and there co ...
The Fundamental Group and Brouwer`s Fixed Point Theorem
... theory. The fundamental group can help answer the question of whether two topological spaces are not homeomorphic. A resulting theorem from studying the fundamental group is the Fixed Point Theorem of Brouwer, which has applications in areas such as economics, game theory, and other fields of math. ...
... theory. The fundamental group can help answer the question of whether two topological spaces are not homeomorphic. A resulting theorem from studying the fundamental group is the Fixed Point Theorem of Brouwer, which has applications in areas such as economics, game theory, and other fields of math. ...
CLASS NOTES MATH 527 (SPRING 2011) WEEK 3 1. Mon, Jan. 31
... where Map(Y, Z) is the space of continuous maps Y −→ Z, equipped with the compact-open topology. But the canonical map Hom(X × Y, Z) −→ Hom(X, Map(Y, Z)) is not surjective for all spaces X, Y , and Z. There are several ways to fix this problem, and the solution we shall take is to work with compactl ...
... where Map(Y, Z) is the space of continuous maps Y −→ Z, equipped with the compact-open topology. But the canonical map Hom(X × Y, Z) −→ Hom(X, Map(Y, Z)) is not surjective for all spaces X, Y , and Z. There are several ways to fix this problem, and the solution we shall take is to work with compactl ...
Chapter VI. Fundamental Group
... Let X be a topological space, x0 its point. A path in X which starts and ends at x0 is a loop in X at x0 . Denote by Ω1 (X, x0 ) the set of loops in X at x0 . Denote by π1 (X, x0 ) the set of homotopy classes of loops in X at x0 . Both Ω1 (X, x0 ) and π1 (X, x0 ) are equipped with a multiplication. ...
... Let X be a topological space, x0 its point. A path in X which starts and ends at x0 is a loop in X at x0 . Denote by Ω1 (X, x0 ) the set of loops in X at x0 . Denote by π1 (X, x0 ) the set of homotopy classes of loops in X at x0 . Both Ω1 (X, x0 ) and π1 (X, x0 ) are equipped with a multiplication. ...
Introduction to higher homotopy groups and
... groups and their uses, and at the same time to prepare a bit for the study of characteristic classes which will come next. These notes are not intended to be a comprehensive reference (most of this material is covered in much greater depth and generality in a number of standard texts), but rather to ...
... groups and their uses, and at the same time to prepare a bit for the study of characteristic classes which will come next. These notes are not intended to be a comprehensive reference (most of this material is covered in much greater depth and generality in a number of standard texts), but rather to ...
Chapter 1: Some Basics in Topology
... points from the boundary. To see this, consider the rectangular representation. We glue the top and bottom edges to the boundary of a disk (the circle). We can imagine that the top edge goes to [0, π] half circle, and bottom edge goes to [π, 2π]. Now imagine that we shrink this rectangle to make its ...
... points from the boundary. To see this, consider the rectangular representation. We glue the top and bottom edges to the boundary of a disk (the circle). We can imagine that the top edge goes to [0, π] half circle, and bottom edge goes to [π, 2π]. Now imagine that we shrink this rectangle to make its ...
STABLE TOPOLOGICAL CYCLIC HOMOLOGY IS TOPOLOGICAL
... It is not surprising that we have to p-complete in the case of TC since the cyclotomic trace is really an invariant of the p-completion of algebraic K-theory, cf. 1.4 below. The rest of this paragraph recalls cyclotomic spectra, topological Hochschild homology, topological cyclic homology and stabil ...
... It is not surprising that we have to p-complete in the case of TC since the cyclotomic trace is really an invariant of the p-completion of algebraic K-theory, cf. 1.4 below. The rest of this paragraph recalls cyclotomic spectra, topological Hochschild homology, topological cyclic homology and stabil ...
Combinatorial Equivalence Versus Topological Equivalence
... which are topologically equivalent but not combinatorially so. See the recent work of Stallings [13] related to this question, also. A weaker question may be asked, which, in the light of Theorem 6.1, is relevant. Are there two combinatorial imbeddings f,g:S,-2_Sm such that these bounded complements ...
... which are topologically equivalent but not combinatorially so. See the recent work of Stallings [13] related to this question, also. A weaker question may be asked, which, in the light of Theorem 6.1, is relevant. Are there two combinatorial imbeddings f,g:S,-2_Sm such that these bounded complements ...
Some theorems concerning absolute neighborhood retracts
... separable metric space such that every homeomorphic image of X as a closed subset of a separable metric space M is a neighborhood retract of M, then X is called an absolute neighborhood retract or an ANR. It is with such spaces that we shall be principally concerned. They are of particular interest ...
... separable metric space such that every homeomorphic image of X as a closed subset of a separable metric space M is a neighborhood retract of M, then X is called an absolute neighborhood retract or an ANR. It is with such spaces that we shall be principally concerned. They are of particular interest ...
About dual cube theorems
... We know by the last theorem that the induced map Gn B → P has a homotopy section s. If ∆n+1 has a lifting through tn , we can consider the map j : B → P induced by the maps idB and B → Tn B. The composite sj is the desired section of gn . ...
... We know by the last theorem that the induced map Gn B → P has a homotopy section s. If ∆n+1 has a lifting through tn , we can consider the map j : B → P induced by the maps idB and B → Tn B. The composite sj is the desired section of gn . ...
Posets and homotopy
... Irreducible points in order theory Beat points are also known in order theory, under the name of irreducible points. In fact, the notion of core was re-discovered by Duffus and Rival (1981). The process of removing irreducible points is called dismantling. The notion has several applications: P ∈ F ...
... Irreducible points in order theory Beat points are also known in order theory, under the name of irreducible points. In fact, the notion of core was re-discovered by Duffus and Rival (1981). The process of removing irreducible points is called dismantling. The notion has several applications: P ∈ F ...
BASIC ALGEBRAIC TOPOLOGY: THE FUNDAMENTAL GROUP OF
... Example 1.4. In R , any two paths f0 and f1 that have the same endpoints are homotopic via the linear homotopoy defined by ft (s) = (1 − t)f0 (s) + tf1 (s). This means that each f0 (s) travels along the line segments to f1 (s) at a constant speed. Proposition 1.5. Given a topological space X with tw ...
... Example 1.4. In R , any two paths f0 and f1 that have the same endpoints are homotopic via the linear homotopoy defined by ft (s) = (1 − t)f0 (s) + tf1 (s). This means that each f0 (s) travels along the line segments to f1 (s) at a constant speed. Proposition 1.5. Given a topological space X with tw ...
from mapping class groups to automorphism groups of free groups
... ∞ → BAut∞ is a map of infinite loop spaces. This means in particular that the map H∗ (Γ∞ ) → H∗ (Aut∞ ) respects the Dyer–Lashof algebra structure. To prove this result, we introduce a new family of groups, the automorphism groups of free groups with boundary, which have the same stable homology as t ...
... ∞ → BAut∞ is a map of infinite loop spaces. This means in particular that the map H∗ (Γ∞ ) → H∗ (Aut∞ ) respects the Dyer–Lashof algebra structure. To prove this result, we introduce a new family of groups, the automorphism groups of free groups with boundary, which have the same stable homology as t ...
MA5209L4 - Maths, NUS - National University of Singapore
... A category C consists of the following three mathematical entities: •A class ob(C), whose elements are called objects; •A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morph ...
... A category C consists of the following three mathematical entities: •A class ob(C), whose elements are called objects; •A class hom(C), whose elements are called morphisms or maps or arrows. Each morphism f has a unique source object a and target object b. We write f: a → b, and we say "f is a morph ...
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.