On Simplicial Loops and H-Spaces 1 Introduction
... The loop space of a space Y or simplicial set Y can be formed in various ways, all equivalent up to homotopy: The usual mapping space Y is an H-group, or better, an A1-space [12], [1]. Using Moore loops, the loop multiplication can be made strictly associative. For a simplicial set Y, there exist ...
... The loop space of a space Y or simplicial set Y can be formed in various ways, all equivalent up to homotopy: The usual mapping space Y is an H-group, or better, an A1-space [12], [1]. Using Moore loops, the loop multiplication can be made strictly associative. For a simplicial set Y, there exist ...
Homology stratifications and intersection homology Geometry & Topology Monographs Colin Rourke Brian Sanderson
... singular cycles (P, f ), (P 0 , f 0 ) is a cycle Q with boundary isomorphic to P ∪−P 0 such that F |P = f , F |P 0 = f 0 . It is well known that (singular) homology can be described as geometric singular homology classes of geometric singular cycles. There is a similar description for relative singu ...
... singular cycles (P, f ), (P 0 , f 0 ) is a cycle Q with boundary isomorphic to P ∪−P 0 such that F |P = f , F |P 0 = f 0 . It is well known that (singular) homology can be described as geometric singular homology classes of geometric singular cycles. There is a similar description for relative singu ...
PDF version - University of Warwick
... singular cycles (P, f ), (P 0 , f 0 ) is a cycle Q with boundary isomorphic to P ∪−P 0 such that F |P = f , F |P 0 = f 0 . It is well known that (singular) homology can be described as geometric singular homology classes of geometric singular cycles. There is a similar description for relative singu ...
... singular cycles (P, f ), (P 0 , f 0 ) is a cycle Q with boundary isomorphic to P ∪−P 0 such that F |P = f , F |P 0 = f 0 . It is well known that (singular) homology can be described as geometric singular homology classes of geometric singular cycles. There is a similar description for relative singu ...
Sheaf Theory (London Mathematical Society Lecture Note Series)
... Sheaf theory provides a language for the discussion of geometric objects of many different kinds. At present it finds its main applications in topology and (more especially) in modern algebraic geometry, where it has been used with great success as a tool in the solution of several longstanding prob ...
... Sheaf theory provides a language for the discussion of geometric objects of many different kinds. At present it finds its main applications in topology and (more especially) in modern algebraic geometry, where it has been used with great success as a tool in the solution of several longstanding prob ...
Elliptic spectra, the Witten genus, and the theorem of the cube
... refinement of the Witten genus, and its modular invariance (1.3), an expression of the modular invariance of the Witten genus of a family. All of this makes it clear that one can deduce special properties of the Witten genus by taking special choices of E. But it also suggests that the really natura ...
... refinement of the Witten genus, and its modular invariance (1.3), an expression of the modular invariance of the Witten genus of a family. All of this makes it clear that one can deduce special properties of the Witten genus by taking special choices of E. But it also suggests that the really natura ...
NON-SPLIT REDUCTIVE GROUPS OVER Z Brian
... discriminant. Thus, by Minkowski’s theorem that every number field K 6= Q has a ramified prime, Spec(Z) has no nontrivial connected finite étale covers. Hence, π1 (Spec(Z)) = 1, so all Z-tori are split. Every Chevalley group G is a Z-model of its split connected reductive generic fiber over Q, and ...
... discriminant. Thus, by Minkowski’s theorem that every number field K 6= Q has a ramified prime, Spec(Z) has no nontrivial connected finite étale covers. Hence, π1 (Spec(Z)) = 1, so all Z-tori are split. Every Chevalley group G is a Z-model of its split connected reductive generic fiber over Q, and ...
THE INVERSE PROBLEM OF GALOIS THEORY 1. Introduction Let
... associated embedding problems of the first and second kind. The obstructions are interpreted as elements of the groups H 1 , H 2 , Ext1 and Ext2 . This brought a number of new results or new proofs of well-known facts, e.g. the second Kochendörffer reduction theorem, which states that every embeddi ...
... associated embedding problems of the first and second kind. The obstructions are interpreted as elements of the groups H 1 , H 2 , Ext1 and Ext2 . This brought a number of new results or new proofs of well-known facts, e.g. the second Kochendörffer reduction theorem, which states that every embeddi ...
Spectra of Small Categories and Infinite Loop Space Machines
... categories Oi and Iα j Oi where α j ∈ α for some finite subset α ⊂ N. Suppose also that the maps of the diagram are defined in a way that allows us to replace the cylinders Iα j for bigger ones Iα j in a natural way. The colimit of the new diagram where the old cylinders were replaced by bigger one ...
... categories Oi and Iα j Oi where α j ∈ α for some finite subset α ⊂ N. Suppose also that the maps of the diagram are defined in a way that allows us to replace the cylinders Iα j for bigger ones Iα j in a natural way. The colimit of the new diagram where the old cylinders were replaced by bigger one ...
Examples - Stacks Project
... coordinate axes) on the affine plane. Then let X1 be the (reduced) full preimage of X0 on the blow-up of the plane (X1 has three rational components forming a chain). Then blow up the resulting surface at the two singularities of X1 , and let X2 be the reduced preimage of X1 (which has five rational ...
... coordinate axes) on the affine plane. Then let X1 be the (reduced) full preimage of X0 on the blow-up of the plane (X1 has three rational components forming a chain). Then blow up the resulting surface at the two singularities of X1 , and let X2 be the reduced preimage of X1 (which has five rational ...
Towards a p-adic theory of harmonic weak Maass forms
... Remark 1.10. Since φ1 and φ2 are regular at all but finitely many points of X the sum on the right is finite. Also, a different choice of local antiderivative for φ1 yields the same pairing, since any two antiderivatives differ by a constant and resP (φ2P ) = 0. Therefore the pairing is well-defined ...
... Remark 1.10. Since φ1 and φ2 are regular at all but finitely many points of X the sum on the right is finite. Also, a different choice of local antiderivative for φ1 yields the same pairing, since any two antiderivatives differ by a constant and resP (φ2P ) = 0. Therefore the pairing is well-defined ...
Chapter 7: Infinite abelian groups For infinite abelian
... Let I = N. • Let A ⊆ N. FA = {B ⊆ N : A ⊆ B}. FA is an ultrafilter. • A filter F on a set I is principal if it is of the form FA = {B ⊆ N : A ⊆ B}. for some subset A ⊆ I. Every principal filter is an ultrafilter. • F1 be the family of co-finite sets. F1 is a non-principal filter but not an ultrafilt ...
... Let I = N. • Let A ⊆ N. FA = {B ⊆ N : A ⊆ B}. FA is an ultrafilter. • A filter F on a set I is principal if it is of the form FA = {B ⊆ N : A ⊆ B}. for some subset A ⊆ I. Every principal filter is an ultrafilter. • F1 be the family of co-finite sets. F1 is a non-principal filter but not an ultrafilt ...
Structured Stable Homotopy Theory and the Descent Problem for
... Thus, the K-theory spectrum of any algebraically closed field is equivalent to the connective complex K-theory spectrum away from the characteristic of the field. The key question now is how to obtain information about the K-theory of a field which is not algebraically closed. An attempt to do this ...
... Thus, the K-theory spectrum of any algebraically closed field is equivalent to the connective complex K-theory spectrum away from the characteristic of the field. The key question now is how to obtain information about the K-theory of a field which is not algebraically closed. An attempt to do this ...
Rings and modules
... Examples. Every abelian group is a Z -module, so the class of abelian groups coincide with the class of Z -modules. Every vector space over a field F is an F -module. 2.2. A map f : M → N is called a homomorphism of A -modules if f (x + y) = f (x) + f (y) for every x, y ∈ M and f (ax) = af (x) for e ...
... Examples. Every abelian group is a Z -module, so the class of abelian groups coincide with the class of Z -modules. Every vector space over a field F is an F -module. 2.2. A map f : M → N is called a homomorphism of A -modules if f (x + y) = f (x) + f (y) for every x, y ∈ M and f (ax) = af (x) for e ...
A Relative Spectral Sequence for Topological Hochschild Homology
... When C is commutative and M is a C ∧A C op - or a C ∧B C op -algebra, respectively, these spectral sequences are multiplicative with respect to the standard products on Hochschild homology and topological Hochschild homology. The method of constructing these spectral sequences is an elaboration of ...
... When C is commutative and M is a C ∧A C op - or a C ∧B C op -algebra, respectively, these spectral sequences are multiplicative with respect to the standard products on Hochschild homology and topological Hochschild homology. The method of constructing these spectral sequences is an elaboration of ...
Notes on étale cohomology
... HomTop(X) (X 0 , ·) of morphisms from the final object X 0 = HomXtop (·, X), and the subcategory of abelian sheaves on Xtop is the subcategory of abelian groups in Top(X), where an abelian group in a category C admitting finite products and a final object e is an object G equipped with maps fitting ...
... HomTop(X) (X 0 , ·) of morphisms from the final object X 0 = HomXtop (·, X), and the subcategory of abelian sheaves on Xtop is the subcategory of abelian groups in Top(X), where an abelian group in a category C admitting finite products and a final object e is an object G equipped with maps fitting ...
Abelian group
... Infinite abelian groups Тhe simplest infinite abelian group is the infinite cyclic group Z. Any finitely generated abelian group A is isomorphic to the direct sum of r copies of Z and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of primary or ...
... Infinite abelian groups Тhe simplest infinite abelian group is the infinite cyclic group Z. Any finitely generated abelian group A is isomorphic to the direct sum of r copies of Z and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of primary or ...
universal covering spaces and fundamental groups in algebraic
... The motivation for gluing together the π1(X, x) (which are individually topological groups) into a group scheme requires some explanation. We wish to study the question: what is a “loop up to homotopy” on a scheme? Grothendieck’s construction of the étale fundamental group gives the beautiful perspe ...
... The motivation for gluing together the π1(X, x) (which are individually topological groups) into a group scheme requires some explanation. We wish to study the question: what is a “loop up to homotopy” on a scheme? Grothendieck’s construction of the étale fundamental group gives the beautiful perspe ...
Sans titre
... Due to the possible failure of Kashiwara’s equivalence for RF DX -modules, the trick of considering the graph inclusion ◆g when D = (g) is not enough to ensure localizability for arbitrary D, so we are forced to considering the possibly smaller category of strictly eX -modules along D which are loca ...
... Due to the possible failure of Kashiwara’s equivalence for RF DX -modules, the trick of considering the graph inclusion ◆g when D = (g) is not enough to ensure localizability for arbitrary D, so we are forced to considering the possibly smaller category of strictly eX -modules along D which are loca ...
My notes - Harvard Mathematics Department
... Zl ⊗ Hom(X, Y ) → HomZl (Tl (X), Tl (Y ))Gal obtained from (3) is an isomorphism. Tate’s conjecture is true when k = Fq (as proved by Tate), and when k is a number field (Faltings). The proofs are quite difficult, and depend a lot on the structure of the Picard group of an abelian variety. 4. We wil ...
... Zl ⊗ Hom(X, Y ) → HomZl (Tl (X), Tl (Y ))Gal obtained from (3) is an isomorphism. Tate’s conjecture is true when k = Fq (as proved by Tate), and when k is a number field (Faltings). The proofs are quite difficult, and depend a lot on the structure of the Picard group of an abelian variety. 4. We wil ...
THE PICARD GROUP OF EQUIVARIANT STABLE HOMOTOPY
... G-spaces, and we first characterize the dualizable G-spectra. Here we are comparing the homotopy category HoGT of based G-spaces to the homotopy category HoGS of G-spectra, and we may restrict attention to based G-CW complexes and to G-CW spectra. We write Σ∞ for the suspension G-spectrum functor Ho ...
... G-spaces, and we first characterize the dualizable G-spectra. Here we are comparing the homotopy category HoGT of based G-spaces to the homotopy category HoGS of G-spectra, and we may restrict attention to based G-CW complexes and to G-CW spectra. We write Σ∞ for the suspension G-spectrum functor Ho ...