K - CIS @ UPenn
... K, of dimension d to be realized in Em, the dimension of the “ambient space”, m, must be big enough. For example, there are 2-complexes that can’t be realized in E3 or even in E4. There has to be enough room in order for condition (2) to be satisfied. It is not hard to prove that m = 2d+1 is always s ...
... K, of dimension d to be realized in Em, the dimension of the “ambient space”, m, must be big enough. For example, there are 2-complexes that can’t be realized in E3 or even in E4. There has to be enough room in order for condition (2) to be satisfied. It is not hard to prove that m = 2d+1 is always s ...
MANIFOLDS, COHOMOLOGY, AND SHEAVES
... the partial derivatives ∂ k f /∂r j1 · · · ∂r jk exist on U for all integers k ≥ 1 and all j1 , . . . , jk . A vector-valued function f = (f 1 , . . . , f m ) : U → Rm is smooth if each component f i is smooth on U . In these lectures we use the words “smooth” and “C ∞ ” interchangeably. A topologic ...
... the partial derivatives ∂ k f /∂r j1 · · · ∂r jk exist on U for all integers k ≥ 1 and all j1 , . . . , jk . A vector-valued function f = (f 1 , . . . , f m ) : U → Rm is smooth if each component f i is smooth on U . In these lectures we use the words “smooth” and “C ∞ ” interchangeably. A topologic ...
Tibor Macko
... Proof of (1). The main idea is to translate the statement about the lift of ν into a statement about orientations with respect to L-theory spectra. The main references for this part are [Ran79] and [Ran92, chapter 16]. Orientations. An orientation of a spherical fibration ν : X → BG(k) with respect ...
... Proof of (1). The main idea is to translate the statement about the lift of ν into a statement about orientations with respect to L-theory spectra. The main references for this part are [Ran79] and [Ran92, chapter 16]. Orientations. An orientation of a spherical fibration ν : X → BG(k) with respect ...
REGULAR CONVERGENCE 1. Introduction. The
... space a V° can be interpreted as a pair of points and being ^ 0 is equivalent to the pair's lying in a connected subset. Thus we have a true generalization of the earlier definition. In this type of convergence it is impossible to close up "holes" with r-dimensional boundaries t h a t exist in the m ...
... space a V° can be interpreted as a pair of points and being ^ 0 is equivalent to the pair's lying in a connected subset. Thus we have a true generalization of the earlier definition. In this type of convergence it is impossible to close up "holes" with r-dimensional boundaries t h a t exist in the m ...
The bordism version of the h
... given a proper solution g : M → N , there exists a smooth triangulation of N such that g is transverse to each simplex ∆ of the triangulation. Choose an order on the set of vertices of the triangulation. Then each simplex ∆ ⊂ N together with g|g−1 ∆ has a unique counterpart in MR and therefore there ...
... given a proper solution g : M → N , there exists a smooth triangulation of N such that g is transverse to each simplex ∆ of the triangulation. Choose an order on the set of vertices of the triangulation. Then each simplex ∆ ⊂ N together with g|g−1 ∆ has a unique counterpart in MR and therefore there ...
Background notes
... These notes, written for another class, are provided for reference. I begin with fiber bundles. Then I will discuss the particular case of vector bundles and the construction of the tangent bundle. Intuitively, the tangent bundle is the disjoint union of the tangent spaces (see (20)). What we must d ...
... These notes, written for another class, are provided for reference. I begin with fiber bundles. Then I will discuss the particular case of vector bundles and the construction of the tangent bundle. Intuitively, the tangent bundle is the disjoint union of the tangent spaces (see (20)). What we must d ...
Frobenius algebras and 2D topological quantum field theories (short
... X −→ Y −→ Z we denote the composition f g. Similarly, we put the symbol of a function to the right of its argument, writing for example f : X −→ Y x 7−→ xf. ...
... X −→ Y −→ Z we denote the composition f g. Similarly, we put the symbol of a function to the right of its argument, writing for example f : X −→ Y x 7−→ xf. ...
Collated Notes on TQFT.pdf
... is categorical language. However, we already need a fair amount of categorical language to explain the first two. The theorem is a comparison of apples and oranges, and it says that these apples and oranges are in some sense the same. We will start with the categorical language that makes sense of s ...
... is categorical language. However, we already need a fair amount of categorical language to explain the first two. The theorem is a comparison of apples and oranges, and it says that these apples and oranges are in some sense the same. We will start with the categorical language that makes sense of s ...
The inverse map of a continuous bijective map might not be
... Fact: Let X�and Y be two topological spaces. Assume X is compact and assume that Y is Hausdorff. Let f : X → Y be a continuous map such that f is also bijective. Then f −1 is a continuous map from Y to X. If X�is not assumed to be compact, then for a bijective map f : X → Y , f being continuous canno ...
... Fact: Let X�and Y be two topological spaces. Assume X is compact and assume that Y is Hausdorff. Let f : X → Y be a continuous map such that f is also bijective. Then f −1 is a continuous map from Y to X. If X�is not assumed to be compact, then for a bijective map f : X → Y , f being continuous canno ...
T A G An invariant of link cobordisms
... The proof of Khovanov’s conjecture implies the existence of a family of derived invariants of link cobordisms with the same source and target, which are analogous to the classical Lefschetz numbers of endomorphisms of manifolds. We call these Lefschetz polynomials of link endocobordisms and like the ...
... The proof of Khovanov’s conjecture implies the existence of a family of derived invariants of link cobordisms with the same source and target, which are analogous to the classical Lefschetz numbers of endomorphisms of manifolds. We call these Lefschetz polynomials of link endocobordisms and like the ...
from mapping class groups to automorphism groups of free groups
... Let h : (∂D × I) × I → X be the projection of this last homotopy and extend it to h∂ : ((∂X × I) ∪ (X × {0, 1})) × I → X using the constant map on ∂X\∂D2 and f and g on (X × 0) × I and (X × 1) × I. Hence h∂ is a homotopy from the restriction H∂ of H to (∂X × I)∪(X × {0, 1}) to the map i∂ X ∪f ∪g : ( ...
... Let h : (∂D × I) × I → X be the projection of this last homotopy and extend it to h∂ : ((∂X × I) ∪ (X × {0, 1})) × I → X using the constant map on ∂X\∂D2 and f and g on (X × 0) × I and (X × 1) × I. Hence h∂ is a homotopy from the restriction H∂ of H to (∂X × I)∪(X × {0, 1}) to the map i∂ X ∪f ∪g : ( ...
Manifolds
... A family At , t ∈ I of subsets of a topological space X is called an isotopy of the set A = A0 , if the graph Γ = {(x, t) ∈ X × I | x ∈ At } of the family is fibrewise homeomorphic to the cylinder A × I, i. e. there exists a homeomorphism A × I → Γ mapping A × {t} to Γ ∩ X × {t} for any t ∈ I. Such ...
... A family At , t ∈ I of subsets of a topological space X is called an isotopy of the set A = A0 , if the graph Γ = {(x, t) ∈ X × I | x ∈ At } of the family is fibrewise homeomorphic to the cylinder A × I, i. e. there exists a homeomorphism A × I → Γ mapping A × {t} to Γ ∩ X × {t} for any t ∈ I. Such ...
IV.2 Homology
... denoted as Zp ≤ Cp , which is a subgroup of the group of p-chains. In other words, the group of p-cycles is the kernel of the p-th boundary homomorphism, Zp = ker ∂p . Since the chain groups are abelian so are their cycle subgroups. Consider p = 0 as an example. The boundary of every vertex is zero, ...
... denoted as Zp ≤ Cp , which is a subgroup of the group of p-chains. In other words, the group of p-cycles is the kernel of the p-th boundary homomorphism, Zp = ker ∂p . Since the chain groups are abelian so are their cycle subgroups. Consider p = 0 as an example. The boundary of every vertex is zero, ...
