Stable ∞-Categories (Lecture 3)
... Remark 5. To any topological category C (and therefore to any ∞-category) we can associate an ordinary category hC, called the homotopy category of C. It has the same objects, with morphisms given by HomhC (X, Y ) = π0 MapC (X, Y ). Example 6. The collection of finite CW complexes forms a topologica ...
... Remark 5. To any topological category C (and therefore to any ∞-category) we can associate an ordinary category hC, called the homotopy category of C. It has the same objects, with morphisms given by HomhC (X, Y ) = π0 MapC (X, Y ). Example 6. The collection of finite CW complexes forms a topologica ...
A BORDISM APPROACH TO STRING TOPOLOGY 1. Introduction
... - Following the works of D. Burghelea and Z. Fiedorowicz [6] of T. Goodwillie [23] and of J.D.S. Jones [29], we know that the cohomology of free loop spaces is strongly related to Hochschild homology and that the S 1 -equivariant cohomology of free loop spaces is also related to cyclic homology and ...
... - Following the works of D. Burghelea and Z. Fiedorowicz [6] of T. Goodwillie [23] and of J.D.S. Jones [29], we know that the cohomology of free loop spaces is strongly related to Hochschild homology and that the S 1 -equivariant cohomology of free loop spaces is also related to cyclic homology and ...
Mumford`s conjecture - University of Oxford
... In his paper Riemann considers how the complex structure of the surfaces associated to a multi-valued complex function changes when one continuously varies the parameters of the function. He concludes that when the genus of the surface is g ≥ 2 the isomorphism class depends on 3g − 3 complex variabl ...
... In his paper Riemann considers how the complex structure of the surfaces associated to a multi-valued complex function changes when one continuously varies the parameters of the function. He concludes that when the genus of the surface is g ≥ 2 the isomorphism class depends on 3g − 3 complex variabl ...
Categories and functors
... The most obvious examples of categories come under the banner ‘categories of mathematical structures’. Example 1.4 There is a category Set in which the objects are sets and the maps are functions. Similarly: • Top is topological spaces and continuous maps • Gp is groups and homomorphisms • Ab is abe ...
... The most obvious examples of categories come under the banner ‘categories of mathematical structures’. Example 1.4 There is a category Set in which the objects are sets and the maps are functions. Similarly: • Top is topological spaces and continuous maps • Gp is groups and homomorphisms • Ab is abe ...
23 Introduction to homotopy theory
... which is a weak equivalence induces a weak equivalence between topologically enriched categories and therefore a weak equivalence of classifying spaces. For example, the usual inclusion O(n) ! GL(n, R) is a weak equivalence, hence we get an equivalence BO(n) ' BGL(n, R). Let G act on a space X. Taki ...
... which is a weak equivalence induces a weak equivalence between topologically enriched categories and therefore a weak equivalence of classifying spaces. For example, the usual inclusion O(n) ! GL(n, R) is a weak equivalence, hence we get an equivalence BO(n) ' BGL(n, R). Let G act on a space X. Taki ...
AAG, LECTURE 13 If 0 → F 1 → F2 → F3 → 0 is a short exact
... useful for theoretical purposes. But because injective objects are typically huge, they are usually completely useless for computations. Fortunately, one can often compute derived functors using more manageable resolutions. Let F : A → B be a left exact functor. Then an object N of A is called F -ac ...
... useful for theoretical purposes. But because injective objects are typically huge, they are usually completely useless for computations. Fortunately, one can often compute derived functors using more manageable resolutions. Let F : A → B be a left exact functor. Then an object N of A is called F -ac ...
STRATIFIED SPACES TWIGS 1. Introduction These
... from the definition, and a stratification satisfying the axiom of the frontier is called a primary stratification. However, the above definition seems to require the least while still being useful, and so we will use it. 3. Examples There are many examples of such spaces, and in fact this is the mai ...
... from the definition, and a stratification satisfying the axiom of the frontier is called a primary stratification. However, the above definition seems to require the least while still being useful, and so we will use it. 3. Examples There are many examples of such spaces, and in fact this is the mai ...
CLASSIFYING SPACES OF MONOIDS – APPLICATIONS IN
... a small category C which is a monoid generated by a single non-unit, idempotent element. Using this, we present here an alternative proof of the contractibility of D2n . Proposition 9. Let Dn := |skn (N (C))| for n = 0, 1, . . ., where C is the monoid C = hx | x2 = xi. Then D2n is conctractible for ...
... a small category C which is a monoid generated by a single non-unit, idempotent element. Using this, we present here an alternative proof of the contractibility of D2n . Proposition 9. Let Dn := |skn (N (C))| for n = 0, 1, . . ., where C is the monoid C = hx | x2 = xi. Then D2n is conctractible for ...
LECTURE NOTES 4: CECH COHOMOLOGY 1
... We shall prove the following. Proposition 6.1. If the cover U is countable, then the presheaf S ∗ U = CU∗ (U, G) of singular cochains which respect U satisfies these hypotheses. Recall that the method of barycentric subdivision proves that for any cover, CU∗ (U, G) and C ∗ (U, G) are chain homotopy ...
... We shall prove the following. Proposition 6.1. If the cover U is countable, then the presheaf S ∗ U = CU∗ (U, G) of singular cochains which respect U satisfies these hypotheses. Recall that the method of barycentric subdivision proves that for any cover, CU∗ (U, G) and C ∗ (U, G) are chain homotopy ...
Section 7: Manifolds with boundary Review definitions of
... manifold (n-mfd for short) if it is Hausdorff, Second Countable, and every point x ∈ X has a neighborhood that is homeomorphic to Rn or Rn+ . A point that has a neighborhood homeomorphic to Rn+ but not to Rn is called a boundary point. The set of all such points (if any) is called the boundary of X, ...
... manifold (n-mfd for short) if it is Hausdorff, Second Countable, and every point x ∈ X has a neighborhood that is homeomorphic to Rn or Rn+ . A point that has a neighborhood homeomorphic to Rn+ but not to Rn is called a boundary point. The set of all such points (if any) is called the boundary of X, ...
Dualities in Mathematics: Locally compact abelian groups
... Let G be a locally compact group. Then G admits a left Haar measure. This measure is unique up to an overall factor. • G has a left Haar measure iff it has a right Haar measure. • A non-zero Haar measure is positive on all open sets. • µ(G) is finite iff G is compact. ...
... Let G be a locally compact group. Then G admits a left Haar measure. This measure is unique up to an overall factor. • G has a left Haar measure iff it has a right Haar measure. • A non-zero Haar measure is positive on all open sets. • µ(G) is finite iff G is compact. ...
English
... = Πn (X, {x0 } , x0 ), for n ≥ 0. The reader is referred to [10] or [23] for the proofs of the above statements. 2.3. CW-complexes. One of the difficulties for computing homotopy groups is that given two arbitrary topological spaces X and Y , it is difficult to construct any continuous map f : X → Y ...
... = Πn (X, {x0 } , x0 ), for n ≥ 0. The reader is referred to [10] or [23] for the proofs of the above statements. 2.3. CW-complexes. One of the difficulties for computing homotopy groups is that given two arbitrary topological spaces X and Y , it is difficult to construct any continuous map f : X → Y ...
PDF
... Examples The long line (without initial point) is homogeneous, but it is not bihomogeneous as its self-homeomorphisms are all order-preserving. This can be considered a pathological example, as most homogeneous topological spaces encountered in practice are also bihomogeneous. Every topological grou ...
... Examples The long line (without initial point) is homogeneous, but it is not bihomogeneous as its self-homeomorphisms are all order-preserving. This can be considered a pathological example, as most homogeneous topological spaces encountered in practice are also bihomogeneous. Every topological grou ...
ORIENTED INTERSECTION MULTIPLICITIES
... Serre and others, is related to the intersection product of rational equivalence classes of algebraic cycles, which gives the multiplication in the Chow ring of algebraic cycles modulo rational equivalence. One definitive algebraic result proved using the Chow ring is Murthy’s Theorem, that over a d ...
... Serre and others, is related to the intersection product of rational equivalence classes of algebraic cycles, which gives the multiplication in the Chow ring of algebraic cycles modulo rational equivalence. One definitive algebraic result proved using the Chow ring is Murthy’s Theorem, that over a d ...
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 , ...
Topological K-theory: problem set 7
... Problem 1. Show that for all n ≥ 3, πn (S3 ) ∼ = π n ( S2 ). Problem 2. Compute K ∗ (CPn ) as an additive group (the ring structure is more complicated). Problem 3. A prespectrum is a sequence of spaces En with maps φn : En → ΩEn+1 (the difference with a spectrum is that we don’t assume the φn to be ...
... Problem 1. Show that for all n ≥ 3, πn (S3 ) ∼ = π n ( S2 ). Problem 2. Compute K ∗ (CPn ) as an additive group (the ring structure is more complicated). Problem 3. A prespectrum is a sequence of spaces En with maps φn : En → ΩEn+1 (the difference with a spectrum is that we don’t assume the φn to be ...
Qualifying Exam in Topology
... 3. (a) Define the two notions: “homotopy between two maps” and “homotopy equivalence between two topological spaces.” (b) Give an example of topological spaces X and Y that have the same homotopy type but are not homeomorphic. (c) Give an example of (path-connected) topological spaces X and Y that h ...
... 3. (a) Define the two notions: “homotopy between two maps” and “homotopy equivalence between two topological spaces.” (b) Give an example of topological spaces X and Y that have the same homotopy type but are not homeomorphic. (c) Give an example of (path-connected) topological spaces X and Y that h ...
Homology (mathematics)
In mathematics (especially algebraic topology and abstract algebra), homology (in part from Greek ὁμός homos ""identical"") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. See singular homology for a concrete version for topological spaces, or group cohomology for a concrete version for groups.For a topological space, the homology groups are generally much easier to compute than the homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.The original motivation for defining homology groups is the observation that shapes are distinguished by their holes. But because a hole is ""not there"", it is not immediately obvious how to define a hole, or how to distinguish between different kinds of holes. Homology is a rigorous mathematical method for defining and categorizing holes in a shape. As it turns out, subtle kinds of holes exist that homology cannot ""see"" — in which case homotopy groups may be what is needed.