Logic – Homework 4

... “There is an x ∈ M , such that P (x) holds.” or (formally) “∃x ∈ M : P (x)” But the syntax of predicate logic shown in the lecture does not allow a construct like ∃x ∈ M . (a) Find a way of expressing such a proposition in predicate logic. (b) Do the same for the all-quantification: “∀x ∈ M : P (x)” ...

... “There is an x ∈ M , such that P (x) holds.” or (formally) “∃x ∈ M : P (x)” But the syntax of predicate logic shown in the lecture does not allow a construct like ∃x ∈ M . (a) Find a way of expressing such a proposition in predicate logic. (b) Do the same for the all-quantification: “∀x ∈ M : P (x)” ...

PDF

... Homology is the general name for a number of functors from topological spaces to abelian groups (or more generally modules over a fixed ring). It turns out that in most reasonable cases a large number of these (singular homology, cellular homology, Morse homology, simplicial homology) all coincide. ...

... Homology is the general name for a number of functors from topological spaces to abelian groups (or more generally modules over a fixed ring). It turns out that in most reasonable cases a large number of these (singular homology, cellular homology, Morse homology, simplicial homology) all coincide. ...

Axiomatic Approach to Homology Theory Author(s)

... although the proofs of some of the axioms are only implicitly contained in the literature. It is well known that the two homology theories differ for some pairs (X, A). Thus, the axioms do not provide uniqueness for all spaces. The surprising feature of H? and H1 that appears in this development is ...

... although the proofs of some of the axioms are only implicitly contained in the literature. It is well known that the two homology theories differ for some pairs (X, A). Thus, the axioms do not provide uniqueness for all spaces. The surprising feature of H? and H1 that appears in this development is ...

4.2 Simplicial Homology Groups

... any two points. Let ϕ be a permutation of points {p0 , p1 , . . . , pk }. Then (pϕ(0) , pϕ(1) , . . . , pϕ(k) ) = (sign ϕ)(p0 , p1 , . . . , pk ), where sign ϕ is the parity of the permutation ϕ. ...

... any two points. Let ϕ be a permutation of points {p0 , p1 , . . . , pk }. Then (pϕ(0) , pϕ(1) , . . . , pϕ(k) ) = (sign ϕ)(p0 , p1 , . . . , pk ), where sign ϕ is the parity of the permutation ϕ. ...

GEOMETRIC REPRESENTATION THEORY OF THE HILBERT SCHEMES PART I

... According to [N, Section 5], the q-dimension of M equals1 n≥0 q n dim H• (X [n] ) = j≥1 1−q j. An easy way to see this is to use a contractable action of the one-dimensional subtorus T1 ⊂ T on X [n] (e.g. T1 = {tN , tN +1 } for N > n). This yields a cell-decomposition of X [n] with the number of cel ...

... According to [N, Section 5], the q-dimension of M equals1 n≥0 q n dim H• (X [n] ) = j≥1 1−q j. An easy way to see this is to use a contractable action of the one-dimensional subtorus T1 ⊂ T on X [n] (e.g. T1 = {tN , tN +1 } for N > n). This yields a cell-decomposition of X [n] with the number of cel ...

IV.2 Homology

... use them to define homology groups. A p-cycle is a p-chain with empty boundary, ∂c = 0. Since ∂ commutes with addition, we have a group of p-cycles, 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 homomorphis ...

... use them to define homology groups. A p-cycle is a p-chain with empty boundary, ∂c = 0. Since ∂ commutes with addition, we have a group of p-cycles, 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 homomorphis ...

2 Continuous maps. 3 Induced topologies.

... If f : X → Y is a function and U is a topology on X, the topology on Y induced by f is the topology whose open sets are exactly the sets V ⊆ Y , with the property that their inverse image are open sets in X. This topology is denoted f∗ (U). The formula below defines the direct image topology. V ∈ f∗ ...

... If f : X → Y is a function and U is a topology on X, the topology on Y induced by f is the topology whose open sets are exactly the sets V ⊆ Y , with the property that their inverse image are open sets in X. This topology is denoted f∗ (U). The formula below defines the direct image topology. V ∈ f∗ ...

Simplicial Sets - Stanford Computer Graphics

... • Linear-time algorithm to reduce the size of a complex • Can use Gaussian Elimination to compute Homology of simplified complex ...

... • Linear-time algorithm to reduce the size of a complex • Can use Gaussian Elimination to compute Homology of simplified complex ...

arXiv:math/0302340v2 [math.AG] 7 Sep 2003

... be the Verdier dual of the retraction. We obtain two maps: ιY and the comI Y . We can rescale α and assume that position f∗ ◦ Rf∗ (ιX ) ◦ α : ICY [2m] → D these two maps coincide on an open set. We will show that they coincide on the whole X. The conclusion will follow. Proposition 6.1 Suppose Y is ...

... be the Verdier dual of the retraction. We obtain two maps: ιY and the comI Y . We can rescale α and assume that position f∗ ◦ Rf∗ (ιX ) ◦ α : ICY [2m] → D these two maps coincide on an open set. We will show that they coincide on the whole X. The conclusion will follow. Proposition 6.1 Suppose Y is ...

Stability and computation of topological invariants of solids in Rn

... is that lfs vanishes on the boundary of non-smooth objects. Theorems involving lfs do not help on non-smooth objetcs, such as solids with sharp edges. Fortunately, algorithms proved correct in the case of smooth objects, behave relatively well in practice on solids with sharp edges. In [7, 8], the a ...

... is that lfs vanishes on the boundary of non-smooth objects. Theorems involving lfs do not help on non-smooth objetcs, such as solids with sharp edges. Fortunately, algorithms proved correct in the case of smooth objects, behave relatively well in practice on solids with sharp edges. In [7, 8], the a ...

Homology Theory - Section de mathématiques

... Homology theory has been around for about 115 years. It’s founding father was the french mathematician Henri Poincaré who gave a somewhat fuzzy definition of what “homology” should be in 1895. Thirty years later, it was realised by Emmy Noether that abelian groups were the right context to study hom ...

... Homology theory has been around for about 115 years. It’s founding father was the french mathematician Henri Poincaré who gave a somewhat fuzzy definition of what “homology” should be in 1895. Thirty years later, it was realised by Emmy Noether that abelian groups were the right context to study hom ...

1. Introduction 1 2. Simplicial and Singular Intersection Homology 2

... We now define singular intersection homology as the homology of the complex whose chains are a suitable subset of the standard singular homology chains. Fix a perversity p. Definition 2.5. A singular i-simplex σ : ∆i → X is said to be p-allowable, if σ −1 (Xm−k ) ⊆ (i − k + p(k)) − skeleton of∆i . S ...

... We now define singular intersection homology as the homology of the complex whose chains are a suitable subset of the standard singular homology chains. Fix a perversity p. Definition 2.5. A singular i-simplex σ : ∆i → X is said to be p-allowable, if σ −1 (Xm−k ) ⊆ (i − k + p(k)) − skeleton of∆i . S ...

Topological Extensions of Linearly Ordered Groups

... A topological space X is called locally compact if for every element x∈ X there exists open neighbourhood U ( x) such that the closure U ( x) is a compact subset of X . Proposition. Let G be a locally compact linearly ordered + with product topology is a topological topological group. Then BG inver ...

... A topological space X is called locally compact if for every element x∈ X there exists open neighbourhood U ( x) such that the closure U ( x) is a compact subset of X . Proposition. Let G be a locally compact linearly ordered + with product topology is a topological topological group. Then BG inver ...

Topology

... example for existence proofs for periodic solutions for systems of partial differential equations (which is, indeed, very close to Poincaré’s1 original motivation for starting off the subject . . . ), and 3. the insight is growing that topological methods can also be useful directly for application ...

... example for existence proofs for periodic solutions for systems of partial differential equations (which is, indeed, very close to Poincaré’s1 original motivation for starting off the subject . . . ), and 3. the insight is growing that topological methods can also be useful directly for application ...

Homology stratifications and intersection homology Geometry & Topology Monographs Colin Rourke Brian Sanderson

... n–complex K and permutation π ∈ Σn+1 . Permutation homology is a convenient device (implicit in Goresky and MacPherson [2]) for studying intersection homology. We prove that, for a PL manifold, all permutation homology groups are the same as ordinary homology groups. In section 3 we prove that the g ...

... n–complex K and permutation π ∈ Σn+1 . Permutation homology is a convenient device (implicit in Goresky and MacPherson [2]) for studying intersection homology. We prove that, for a PL manifold, all permutation homology groups are the same as ordinary homology groups. In section 3 we prove that the g ...

PDF version - University of Warwick

... n–complex K and permutation π ∈ Σn+1 . Permutation homology is a convenient device (implicit in Goresky and MacPherson [2]) for studying intersection homology. We prove that, for a PL manifold, all permutation homology groups are the same as ordinary homology groups. In section 3 we prove that the g ...

... n–complex K and permutation π ∈ Σn+1 . Permutation homology is a convenient device (implicit in Goresky and MacPherson [2]) for studying intersection homology. We prove that, for a PL manifold, all permutation homology groups are the same as ordinary homology groups. In section 3 we prove that the g ...

Aspherical manifolds that cannot be triangulated

... (a) h(K) is a locally CAT(0) cubical complex; in particular, it is aspherical. (b) The inverse image in h(K) of any simplex of K is a “hyperbolized simplex”. So, the inverse image of each vertex in K is a point in h(K). (c) f : h(K) → K induces a split injection on cohomology (cf. [3, p. 355]). (d) ...

... (a) h(K) is a locally CAT(0) cubical complex; in particular, it is aspherical. (b) The inverse image in h(K) of any simplex of K is a “hyperbolized simplex”. So, the inverse image of each vertex in K is a point in h(K). (c) f : h(K) → K induces a split injection on cohomology (cf. [3, p. 355]). (d) ...