Kyoto University. Note on the cohomology of finite cyclic coverings. Yasuhiro Hara and Daisuke Kishimoto


 Emerald Summers
 2 years ago
 Views:
Transcription
1 Kyoto University KyotoMath Note on the cohomology of finite cyclic coverings by Yasuhiro Hara and Daisuke Kishimoto April 2013 Department of Mathematics Faculty of Science Kyoto University Kyoto , JAPAN
2 NOTE ON THE COHOMOLOGY OF FINITE CYCLIC COVERINGS YASUHIRO HARA AND DAISUKE KISHIMOTO Abstract. We introduce the height of a normal cyclic pfold covering and show a cohomological relation between the base and the total spaces of the covering in terms of the height. 1. Statement of results The purpose of this note is to show a cohomological property of a normal cyclic pfold covering with respect to a certain cuplength type invariant of the covering. Let p be a prime and let E B be a normal cyclic pfold covering where B is path connected. Suppose p = 2. In [Ko], Kozlov defined the height of the covering h(e) as the maximum n such that w 1 (E) n 0, where w 1 (E) is the first StiefelWhitney class of the covering. By a chain level consideration, he proved H h(e) (E; Z/2) 0. This also follows immediately from the Gysin sequence of the double covering E B. We would like to generalize this result to any prime p. Let p be an arbitrary prime. Let C p be a cyclic group of order p and let ρ : B BC p be the classifying map of the covering E B. The height of the covering can be generalized as h(e) = maxn ρ : H n (BC p ; Z/p) H n (B; Z/p) is nontrivial}. We remark here that the height of a normal cyclic pfold covering is closely related with the idealvalued cohomological index theory of Fadell and Husseini [FH1] and hence the BorsukUlam theorem. We will interpret the height in terms of the category weight introduced by Fadell and Husseini [FH2] and studied further by Rudyak [Ru] and Strom [S]. The most difficult point in generalizing the result of Kozlov is the nonexistence of the Gysin sequence for the covering E B when p is odd. However, we define the corresponding spectral sequence by which we prove: Theorem 1.1. Let E B be a normal cyclic pfold covering, where B is pathconnected. Then As an immediate corollary, we have: H h(e) (E; Z/p) 0. Corollary 1.2. Let E B be a normal cyclic pfold covering, where B is pathconnected. If h(e) n and H n (E; Z/p) = 0, it holds that h(e) n + 1. In section 2, we construct a spectral sequence for a normal cyclic pfold covering which calculate the mod p cohomology of the total space from the base space whose differential is shown to be given as a certain higher Massey product of Kraines [Kr]. Using this spectral sequence, we prove 1
3 2 YASUHIRO HARA AND DAISUKE KISHIMOTO Theorem 1.1. In section 3, we interpret the height of a normal cyclic pfold covering in terms of the category weight introduced by Fadell and Husseini [FH2] and elaborated by [Ru] and [S]. Acknowledgement. The authors are grateful to the referee for leading them to the proof using the spectral sequence. In the first version of the paper, the proof is done in a quite elementary but lengthy way using the Smith special cohomology. (cf. [B]) 2. Proof of Theorem 1.1 Throughout this section, let p be an odd prime and the coefficient of cohomology is Z/p Spectral sequence. Let E B be a normal pfold covering where B is pathconnected. In this subsection, we introduce a spectral sequence which calculates the mod p cohomology of E from B. Analogous spectral sequences were considered in [F] and [Re]. We first set notation. Let ρ : B BC p be the classifying map of the covering E B. Recall that the mod p cohomology of BC p is given as H (BC p ) = Λ(u) Z/p[v], βu = v, u = 1, where β is the Bockstein operation. We denote the cohomology classes ρ (u) and ρ (v) of B by ū and v, respectively. Let R[C p ] denote the group ring of C p over a ring R. Note that the singular chain complex S (E) is a free Z[C p ]module. We regard Z/p[C p ] and Z/p as Z[C p ]modules by the modulo p reduction and the trivial C p action, respectively. Then there are natural isomorphisms (2.1) H (Hom Z[Cp](S (E), Z/p[C p ])) = H (E) and H (Hom Z[Cp](S (E), Z/p)) = H (B). We now fix a generator g of C p and put τ = 1 g Z/p[C p ]. Observe that Z/p[C p ] = Z/p[τ]/(τ p ). Consider the filtration 0 τ p 1 Z/p[C p ] τ p 2 Z/p[C p ] τz/p[c p ] Z/p[C p ]. Then there is a spectral sequence (E r, d r ) associated with the induced filtration of the cochain complex Hom Z[Cp ](S (E), Z/p[C p ]). By (2.1), we have (2.2) E s,t 1 = H t (B) 0 s p 1 0 otherwise H (E) and the degree of the differential d r is ( r, 1), where the total degree of Er s,t is t. Let us identify the differential of this spectral sequence. To this end, we calculate the induced coboundary map δ of the associated graded cochain complex p 1 p 1 Hom Z[Cp](S (E), τ i Z/p[C p ]/τ i 1 Z/p[C p ]) = τ i Hom Z (S (B), Z/p). i=0 In the special case of the universal bundle EC p BC p, we may put i=0 δ(1) = τu τ p 1 u p 1, u i Hom Z (S 1 (B), Z/p)
4 NOTE ON THE COHOMOLOGY OF FINITE CYCLIC COVERINGS 3 for 1 Hom Z (S 0 (B), Z/p). Consider the map E ρ π EC p B, where ρ is a lift of ρ and π is the projection. Then we see that (2.3) δx = δx + τρ (u 1 )x + + τ p 1 ρ (u p 1 )x. for any x Hom Z (S (B), Z/p) in general. If [u 1 ] = 0, 1 E 1,0 becomes a permanent cycle in the spectral sequence (2.2) for the universal bundle EC p BC p, which contradicts to the contractibility of EC p. Then by normalizing u if necessary, we may assume (2.4) [u 1 ] = u. Applying (2.3) in turn to u 1,..., u p 1, we inductively see from the equality δ 2 = 0 that (2.5) δu i = j<i u j u i j for i 2. Let x 1,..., x n n stand for the nfold Massey product in the sense of Kraines [Kr], where x 1, x 2 = ±x 1 x 2. Then by (2.3), (2.4) and (2.5), we obtain that d r x is represented by an element of ± ū,..., ū, x r+1 whose defining system x ij } 1 i j r+1 satisfies x ij = ρ (u j i+1 ) for j r, where x i,r+1 can be an arbitrary cochain satisfying the condition of defining systems. Hence by [Kr], x ij } 1 i j r is the pullback of a defining system for 0} k < p (2.6) u,..., u k = v} k = p. Recall the following associativity formula of higher Massey products [May]. Suppose a defining system for x 1,..., x n 1 n 1 extends to those of x k+1,..., x n n k. Put x ij} 1 i j k+1 (2.7) x ij = ±x ij for j k and x i,k+1 = n 1 l=k+1 ±x il x ln for 2 i k + 1. Then x ij} 1 i j k+1 is a defining system for x 1,..., x k, x k+1,..., x n n k k+1 and the resulting element x satisfies x = ±yx n for some y x 1,..., x n 1 n 1. Consider the defining system of ū,..., ū r+r given by ρ (u i ) for r + r p. By the above observation on d r x, we can extend this defining system to that for ū,..., ū, x r +1 as (2.7) so that the resulting element x represents d r x. Moreover, by (2.6) and the above associativity formula, we have (2.8) d r x 0 r + r < p = ± vx r + r = p.
5 4 YASUHIRO HARA AND DAISUKE KISHIMOTO 2.2. Proof of Theorem 1.1. We prove the result by calculating the spectral sequence (2.2). We first consider the case h(e) = 2m + 1. We can easily see that in the spectral sequence for the universal bundle EC p BC p, it holds that d p 1,2m+1 r uv m = 0 and av m+1 according as r < p 1 and r = p 1, where a (Z/p). Then it follows from naturality of the spectral sequence that d p 1,2m+1 r ū v m = ρ (d p 1,2m+1 r uv m 0 r < p 1 ) = ρ (av m+1 ) = 0 r = p 1, implying that H 2m+1 (E) 0. We next consider the case h(e) = 2m. Let r be the maximum integer such that v m E s,2m 1 survives at the E r term for all 0 s p 1. Suppose that d s,2m r v m 0 for some s. Then we have (2.9) d r,2m r v m 0. If v m E r 1,2m 1 survives at the E r term for r r and satisfies d r+r 1,2m 1 r x = v m for some x, we have d r,2m r v m ± ū,..., ū, v m r+1, v m ± ū,..., ū, x r +1 and r + r p, where defining systems for both higher Massey products are described above. Then it follows from (2.8) that (2.10) d r,2m r v m 0 r + r < p = ± vx r + r = p in the E r term. The upper case contradicts to (2.9). Let us consider the lower case. If r = 1, ūx = v and then β(ūx) = 0. If r 2, ūx = 0 and so β(ūx) = 0. Then in both cases, we have vx = ū(βx), and so vx turns out to be trivial in the E r term, which contradicts to (2.9). Therefore we obtain that v m E r 1,2m 1 is a permanent cycle, implying that H 2m (E) 0. Suppose next that dr s,2m 1 x = v m for some s. Then for any r + r p, we can choose a representative of d r+1,2m r v m as above, and hence by an argument similar to the above case, we see that v m E r+1,2m 1 is a permanent cycle, implying that H 2m (E) 0. Therefore the proof of Theorem 1.1 is completed. 3. Height and category weight In this section, we interpret the height of a normal cyclic pfold covering in terms of the category weight introduced by Fadell and Husseini [FH2] and studied further by Rudyak [Ru] and Strom [S]. As a consequence, the relation between the height of a normal cyclic pfold covering and the LusternikSchnirelmann (LS, for short) category of the classifying map of the covering becomes clear. Recall that the LS category of a space X, denoted by cat(x), is the minimum n such that there is a cover of X by (n + 1)open sets each of which is contractible in X. In [BG], the LS category of a space was generalized to a map: The LS category of a map f : X Y, denoted by cat(f), is the minimum integer n such that there exists an open cover X = U 0 U n where the restriction of f to U i is nullhomotopic for all i. Observe that cat(f) cat(1 X ) = cat(x).
6 NOTE ON THE COHOMOLOGY OF FINITE CYCLIC COVERINGS 5 It is useful to evaluate cat(f) by the socalled Ganea spaces. Let G n (Y ) be the n th Ganea space of Y and let π n : G n (Y ) Y be the projection. See [CLOT] for definition. We know that cat(f) n if and only if there is a map g : X G n (Y ) satisfying π n g f. The homotopy invariant version of the category weight of a space X due to Rudyak [Ru] and Strom [S] is a lower bound for the LS category of X which refines the cuplength. As in [IK], cohomologically, the idea of the homotopy invariant version of the category weight due to Rudyak and Strom is summarized as wgt(x; R) = maxn π n : H (X; R) H (G n (X); R) is injective}, where R is a ring and H denotes the reduced cohomology. By definition, wgt(x; R) is bounded above by cat(x). Given a map f : X Y, we can easily generalize the above definition for a space to a map as wgt(f; R) = maxn there exists y H (Y ; R) satisfying f (y) 0, and π n(z) 0 whenever f (z) 0 for z H (Y ; R)}. Notice that wgt(1 X ; R) = wgt(x; R) analogously to the LS category. Obviously, we have cat(f) wgt(f; R). Let us consider the relation between the height of a normal cyclic covering and the category weight. Suppose a space Y is pathconnected. In general, since the homotopy fiber of the projection π n : G n (Y ) Y has the homotopy type of the join of (n + 1)copies of ΩY which is nconnected, the induced map π n : H k (Y ; R) H k (G n (Y ); R) is an isomorphism for k < n and is injective for k = n. See [CLOT]. We specialize to the case Y = BC p. Recall that G n (BC p ) has the homotopy type of the quotient of the join of the (n + 1)copies of C p by the diagonal free C p  action, implying that G n (BC p ) has the homotopy type of an ndimensional CWcomplex. Then the induced map π n : H k (BC p ; R) H k (G n (BC p ); R) is the zero map for k > n. Summarizing, the induced map π n : H k (BC p ; Z/p) H k (G n (BC p ); Z/p) is injective for k n and is the zero map for k > n, and hence for a map f : X BC p, we have Therefore we obtain: wgt(f; Z/p) = minn f : H n (BC p ; Z/p) H n (X; Z/p) is nontrivial}. Proposition 3.1. Let E B be a normal cyclic pfold covering with the classifying map ρ : B BC p, where B is pathconnected. Then h(e) = wgt(ρ; Z/p) cat(ρ) cat(b). References [B] G.E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics 46, Academic Press, New YorkLondon, [BG] I. Berstein and T. Ganea, The category of a map and of a cohomology class, Fund. Math. 50 (1961/1962), [CLOT] O. Cornea, G. Lupton, J. Oprea and D. Tanré, LusternikSchnirelmann category, Mathematical Surveys and Monographs 103, Ame. Math. Soc., Providence, RI, 2003.
7 6 YASUHIRO HARA AND DAISUKE KISHIMOTO [F] M. Farber, Topology of closed oneforms, Math. Surveys Monogr., vol. 108, Amer. Math. Soc., Providence, RI, [FH1] E. Fadell and S. Husseini, An idealvalued cohomologiucal index theory with applications to BorsukUlam and BourginYang theorem, Ergodic Theory Dynam. System 8 (1988), Charles Conely Memorial Issue, [FH2] E. Fadell and S. Husseini, Category weight and Steenrod operations, Papers in honor os José Adem (Spanish). Bol. Soc. Mat. Mexicana (2) 37 (1992), no.12, [IK] N. Iwase and A. Kono, LusternikSchnirelmann category of Spin(9), Trans. Amer. Math. Soc., 359 (2007), [Ko] D.N. Kozlov, Homology tests for graph colorings, Algebraic and geometric combinatorics, , Contemp. Math. 423, Amer. Math. Soc. Providence, RI, [Kr] D. Kraines, Massey higher products, Trans. Amer. Math. Soc. 124 (1966) [May] J.P. May, Matric Massey products, J. Algebra 12 (1969) [Mat] J. Matoušek, Using the BorsukUlam theorem, Lectures on Topological Methods in Combinatorics and Geometry, Universitext, SpringerVerlag, Heidelberg, [Re] A. Reznikov, Threemanifolds class field theory (homology of coverings for a nonvirtually b 1 positive manifold), Selecta Math. (N.S.) 3 (1997), no. 3, [Ru] Y.B. Rudyak, On category weight and its applications, Topology 38 (1999), no. 1, [S] J. Strom, Essential category weight and phantom maps, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, , Japan address: Department of Mathematics, Kyoto University, Kyoto, , Japan address:
On the existence of Gequivariant maps
CADERNOS DE MATEMÁTICA 12, 69 76 May (2011) ARTIGO NÚMERO SMA# 345 On the existence of Gequivariant maps Denise de Mattos * Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação,
More informationMath 231b Lecture 35. G. Quick
Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the Jhomomorphism could be defined by
More informationHomological Algebra Workshop 020315/060315
Homological Algebra Workshop 020315/060315 Monday: Tuesday: 9:45 10:00 Registration 10:00 11:15 Conchita Martínez (I) 10:00 11:15 Conchita Martínez (II) 11:15 11:45 Coffee Break 11:15 11:45 Coffee
More informationFIXED POINT SETS OF FIBERPRESERVING MAPS
FIXED POINT SETS OF FIBERPRESERVING MAPS Robert F. Brown Department of Mathematics University of California Los Angeles, CA 90095 email: rfb@math.ucla.edu Christina L. Soderlund Department of Mathematics
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationSets of Fibre Homotopy Classes and Twisted Order Parameter Spaces
Communications in Mathematical Physics ManuscriptNr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan BechtluftSachs, Marco Hien Naturwissenschaftliche
More informationFIBER BUNDLES AND UNIVALENCE
FIBER BUNDLES AND UNIVALENCE TALK BY IEKE MOERDIJK; NOTES BY CHRIS KAPULKIN This talk presents a proof that a universal Kan fibration is univalent. The talk was given by Ieke Moerdijk during the conference
More informationThe fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992), 33 37) Bart de Smit
The fundamental group of the Hawaiian earring is not free Bart de Smit The fundamental group of the Hawaiian earring is not free (International Journal of Algebra and Computation Vol. 2, No. 1 (1992),
More informationA NOTE ON TRIVIAL FIBRATIONS
A NOTE ON TRIVIAL FIBRATIONS Petar Pavešić Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani, Jadranska 19, 1111 Ljubljana, Slovenija petar.pavesic@fmf.unilj.si Abstract We study the conditions
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationA homotopytheoretic view of Bott Taubes integrals
Stanford University November 8, 2009 Intro We will construct cohomology classes in spaces of knots K where K = Emb(S 1, R n ) or Emb(R, R n ) (n 3). Connected components of K correspond to isotopy classes
More informationBPcohomology of mapping spaces from the classifying space of a torus to some ptorsion free space
BPcohomology of mapping spaces from the classifying space of a torus to some ptorsion free space 1. Introduction Let p be a fixed prime number, V a group isomorphic to (Z/p) d for some integer d and
More informationAllen Back. Oct. 29, 2009
Allen Back Oct. 29, 2009 Notation:(anachronistic) Let the coefficient ring k be Q in the case of toral ( (S 1 ) n) actions and Z p in the case of Z p tori ( (Z p )). Notation:(anachronistic) Let the coefficient
More informationFiber bundles and nonabelian cohomology
Fiber bundles and nonabelian cohomology Nicolas Addington April 22, 2007 Abstract The transition maps of a fiber bundle are often said to satisfy the cocycle condition. If we take this terminology seriously
More informationRIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES
RIGIDITY OF HOLOMORPHIC MAPS BETWEEN FIBER SPACES GAUTAM BHARALI AND INDRANIL BISWAS Abstract. In the study of holomorphic maps, the term rigidity refers to certain types of results that give us very specific
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationEVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY MITSUNOBU TSUTAYA
EVALUATION FIBER SEQUENCE AND HOMOTOPY COMMUTATIVITY MITSUNOBU TSUTAYA Abstract. We review the relation between the evaluation fiber sequence of mapping spaces and the various higher homotopy commutativities.
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationAnalytic cohomology groups in top degrees of Zariski open sets in P n
Analytic cohomology groups in top degrees of Zariski open sets in P n Gabriel Chiriacescu, Mihnea Colţoiu, Cezar Joiţa Dedicated to Professor Cabiria Andreian Cazacu on her 80 th birthday 1 Introduction
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An ndimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0534405967. Systems of Linear Equations Definition. An ndimensional vector is a row or a column
More informationSystems of Linear Equations
Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and
More informationChapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring
More informationTHE HOPF RING FOR COMPLEX COBORDISM 1 BY DOUGLAS C. RAVENEL AND W. STEPHEN WILSON. Communicated by Edgar Brown, Jr., May 13, 1974
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 6, November 1974 THE HOPF RING FOR COMPLEX COBORDISM 1 BY DOUGLAS C. RAVENEL AND W. STEPHEN WILSON Communicated by Edgar Brown, Jr., May
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationCyclotomic Extensions
Chapter 7 Cyclotomic Extensions A cyclotomic extension Q(ζ n ) of the rationals is formed by adjoining a primitive n th root of unity ζ n. In this chapter, we will find an integral basis and calculate
More informationA CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE
A CONSTRUCTION OF THE UNIVERSAL COVER AS A FIBER BUNDLE DANIEL A. RAMRAS In these notes we present a construction of the universal cover of a path connected, locally path connected, and semilocally simply
More informationAN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC
AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l4] on graphs and matroids I used definitions equivalent to the following.
More informationDIVISORS AND LINE BUNDLES
DIVISORS AND LINE BUNDLES TONY PERKINS 1. Cartier divisors An analytic hypersurface of M is a subset V M such that for each point x V there exists an open set U x M containing x and a holomorphic function
More informationMATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.
MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted
More informationPOLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS
POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationRINGS WITH A POLYNOMIAL IDENTITY
RINGS WITH A POLYNOMIAL IDENTITY IRVING KAPLANSKY 1. Introduction. In connection with his investigation of projective planes, M. Hall [2, Theorem 6.2]* proved the following theorem: a division ring D in
More informationHatcher C A: [012] = [013], B : [123] = [023],
Ex 2..2 Hatcher 2. Let S = [2] [23] 3 = [23] be the union of two faces of the 3simplex 3. Let be the equivalence relation that identifies [] [3] and [2] [23]. The quotient space S/ is the Klein bottle
More informationCycles in a Graph Whose Lengths Differ by One or Two
Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDEBERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS
More informationA NEW CONSTRUCTION OF 6MANIFOLDS
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 8, August 2008, Pages 4409 4424 S 00029947(08)044620 Article electronically published on March 12, 2008 A NEW CONSTRUCTION OF 6MANIFOLDS
More informationAlgebraic KTheory of Ring Spectra (Lecture 19)
Algebraic KTheory of ing Spectra (Lecture 19) October 17, 2014 Let be an associative ring spectrum (here by associative we mean A or associative up to coherent homotopy; homotopy associativity is not
More informationFIBER PRODUCTS AND ZARISKI SHEAVES
FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also
More information8.1 Examples, definitions, and basic properties
8 De Rham cohomology Last updated: May 21, 211. 8.1 Examples, definitions, and basic properties A kform ω Ω k (M) is closed if dω =. It is exact if there is a (k 1)form σ Ω k 1 (M) such that dσ = ω.
More informationOnline publication date: 11 March 2010 PLEASE SCROLL DOWN FOR ARTICLE
This article was downloaded by: [Modoi, George Ciprian] On: 12 March 2010 Access details: Access Details: [subscription number 919828804] Publisher Taylor & Francis Informa Ltd Registered in England and
More informationON FIBER DIAMETERS OF CONTINUOUS MAPS
ON FIBER DIAMETERS OF CONTINUOUS MAPS PETER S. LANDWEBER, EMANUEL A. LAZAR, AND NEEL PATEL Abstract. We present a surprisingly short proof that for any continuous map f : R n R m, if n > m, then there
More informationF. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)
Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationADDITIVE GROUPS OF RINGS WITH IDENTITY
ADDITIVE GROUPS OF RINGS WITH IDENTITY SIMION BREAZ AND GRIGORE CĂLUGĂREANU Abstract. A ring with identity exists on a torsion Abelian group exactly when the group is bounded. The additive groups of torsionfree
More information4. CLASSES OF RINGS 4.1. Classes of Rings class operator Aclosed Example 1: product Example 2:
4. CLASSES OF RINGS 4.1. Classes of Rings Normally we associate, with any property, a set of objects that satisfy that property. But problems can arise when we allow sets to be elements of larger sets
More informationON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE
Illinois Journal of Mathematics Volume 46, Number 1, Spring 2002, Pages 145 153 S 00192082 ON THE NUMBER OF REAL HYPERSURFACES HYPERTANGENT TO A GIVEN REAL SPACE CURVE J. HUISMAN Abstract. Let C be a
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationFIBRATION SEQUENCES AND PULLBACK SQUARES. Contents. 2. Connectivity and fiber sequences. 3
FIRTION SEQUENES ND PULLK SQURES RY MLKIEWIH bstract. We lay out some foundational facts about fibration sequences and pullback squares of topological spaces. We pay careful attention to connectivity ranges
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationA simple criterion on degree sequences of graphs
Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree
More information11 Ideals. 11.1 Revisiting Z
11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(
More informationSubsets of Euclidean domains possessing a unique division algorithm
Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness
More informationDiagonal, Symmetric and Triangular Matrices
Contents 1 Diagonal, Symmetric Triangular Matrices 2 Diagonal Matrices 2.1 Products, Powers Inverses of Diagonal Matrices 2.1.1 Theorem (Powers of Matrices) 2.2 Multiplying Matrices on the Left Right by
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationLecture 4 Cohomological operations on theories of rational type.
Lecture 4 Cohomological operations on theories of rational type. 4.1 Main Theorem The Main Result which permits to describe operations from a theory of rational type elsewhere is the following: Theorem
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 918/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationIntroduction to Finite Fields (cont.)
Chapter 6 Introduction to Finite Fields (cont.) 6.1 Recall Theorem. Z m is a field m is a prime number. Theorem (Subfield Isomorphic to Z p ). Every finite field has the order of a power of a prime number
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More information10. Graph Matrices Incidence Matrix
10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a
More informationThe Open University s repository of research publications and other research outputs
Open Research Online The Open University s repository of research publications and other research outputs The degreediameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:
More informationSymplectic structures on fiber bundles
Symplectic structures on fiber bundles François Lalonde Université du Québec à Montréal (flalonde@math.uqam.ca) Dusa McDuff State University of New York at Stony Brook (dusa@math.sunysb.edu) Aug 27, 2000,
More informationNonzero degree tangential maps between dual symmetric spaces
ISSN 14722739 (online) 14722747 (printed) 709 Algebraic & Geometric Topology Volume 1 (2001) 709 718 Published: 30 November 2001 ATG Nonzero degree tangential maps between dual symmetric spaces Boris
More informationInvariant Metrics with Nonnegative Curvature on Compact Lie Groups
Canad. Math. Bull. Vol. 50 (1), 2007 pp. 24 34 Invariant Metrics with Nonnegative Curvature on Compact Lie Groups Nathan Brown, Rachel Finck, Matthew Spencer, Kristopher Tapp and Zhongtao Wu Abstract.
More informationSMOOTH CASE FOR T or Z
SMOOTH CASE FOR T or Z Preliminaries. Theorem [DDK] Let F be a simply finite connected CWcomplex and E 0 (F ) be the connected component of identity of the space of all homotopy equivalences F F. Then
More informationNOTES ON CATEGORIES AND FUNCTORS
NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category
More informationAXIOMS FOR INVARIANT FACTORS*
PORTUGALIAE MATHEMATICA Vol 54 Fasc 3 1997 AXIOMS FOR INVARIANT FACTORS* João Filipe Queiró Abstract: We show that the invariant factors of matrices over certain types of rings are characterized by a short
More informationDoug Ravenel. October 15, 2008
Doug Ravenel University of Rochester October 15, 2008 s about Euclid s Some s about primes that every mathematician should know (Euclid, 300 BC) There are infinitely numbers. is very elementary, and we
More informationModule MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013
Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationTHE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP
THE AVERAGE DEGREE OF AN IRREDUCIBLE CHARACTER OF A FINITE GROUP by I. M. Isaacs Mathematics Department University of Wisconsin 480 Lincoln Dr. Madison, WI 53706 USA EMail: isaacs@math.wisc.edu Maria
More informationWeek 5: Binary Relations
1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all
More informationON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER
Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group
More informationTOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS
TOPOLOGY OF SINGULAR FIBERS OF GENERIC MAPS OSAMU SAEKI Dedicated to Professor Yukio Matsumoto on the occasion of his 60th birthday Abstract. We classify singular fibers of C stable maps of orientable
More informationRESEARCH STATEMENT AMANDA KNECHT
RESEARCH STATEMENT AMANDA KNECHT 1. Introduction A variety X over a field K is the vanishing set of a finite number of polynomials whose coefficients are elements of K: X := {(x 1,..., x n ) K n : f i
More informationThe Australian Journal of Mathematical Analysis and Applications
The Australian Journal of Mathematical Analysis and Applications Volume 7, Issue, Article 11, pp. 114, 011 SOME HOMOGENEOUS CYCLIC INEQUALITIES OF THREE VARIABLES OF DEGREE THREE AND FOUR TETSUYA ANDO
More informationSign changes of Hecke eigenvalues of Siegel cusp forms of degree 2
Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationSmith Theory Revisited
Smith Theory Revisited William G. Dwyer Clarence W. Wilkerson 1 Introduction In the late 1930 s, P. A. Smith began the investigation of the cohomological properties of a group G of prime order p acting
More informationINVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS
INVARIANT METRICS WITH NONNEGATIVE CURVATURE ON COMPACT LIE GROUPS NATHAN BROWN, RACHEL FINCK, MATTHEW SPENCER, KRISTOPHER TAPP, AND ZHONGTAO WU Abstract. We classify the leftinvariant metrics with nonnegative
More informationSurvey and remarks on Viro s definition of Khovanov homology
RIMS Kôkyûroku Bessatsu B39 (203), 009 09 Survey and remarks on Viro s definition of Khovanov homology By Noboru Ito Abstract This paper reviews and offers remarks upon Viro s definition of the Khovanov
More informationNONCANONICAL EXTENSIONS OF ERDŐSGINZBURGZIV THEOREM 1
NONCANONICAL EXTENSIONS OF ERDŐSGINZBURGZIV THEOREM 1 R. Thangadurai StatMath Division, Indian Statistical Institute, 203, B. T. Road, Kolkata 700035, INDIA thanga v@isical.ac.in Received: 11/28/01,
More informationTwo classes of ternary codes and their weight distributions
Two classes of ternary codes and their weight distributions Cunsheng Ding, Torleiv Kløve, and Francesco Sica Abstract In this paper we describe two classes of ternary codes, determine their minimum weight
More informationCOFINAL MAXIMAL CHAINS IN THE TURING DEGREES
COFINA MAXIMA CHAINS IN THE TURING DEGREES WEI WANG, IUZHEN WU, AND IANG YU Abstract. Assuming ZF C, we prove that CH holds if and only if there exists a cofinal maximal chain of order type ω 1 in the
More informationON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP. A. K. Das and R. K. Nath
International Electronic Journal of Algebra Volume 7 (2010) 140151 ON GENERALIZED RELATIVE COMMUTATIVITY DEGREE OF A FINITE GROUP A. K. Das and R. K. Nath Received: 12 October 2009; Revised: 15 December
More informationGEOMETRIC PROPERTIES OF PROJECTIVE MANIFOLDS OF SMALL DEGREE
GEOMETRIC PROPERTIES OF PROJECTIVE MANIFOLDS OF SMALL DEGREE SIJONG KWAK AND JINHYUNG PARK Abstract. We study geometric structures of smooth projective varieties of small degree in birational geometric
More informationAlgebraic Morava Ktheories and the. higher degree formula
Algebraic Morava Ktheories and the higher degree formula 1 Contents Chapter 1. Introduction 1 Chapter 2. The construction of an algebraic Morava Ktheory spectrum 5 Chapter 3. The theorem in the general
More informationarxiv:1209.1368v1 [math.sg] 6 Sep 2012
ISOTOPY CLASSIFICATION OF ENGEL STRUCTURES ON CIRCLE BUNDLES MIRKO KLUKAS AND BIJAN SAHAMIE arxiv:1209.1368v1 [math.sg] 6 Sep 2012 Abstract. We call two Engel structures isotopic if they are homotopic
More informationSUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by
SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples
More informationIntroduction to Topology
Introduction to Topology Tomoo Matsumura November 30, 2010 Contents 1 Topological spaces 3 1.1 Basis of a Topology......................................... 3 1.2 Comparing Topologies.......................................
More informationThe Topology of Fiber Bundles Lecture Notes. Ralph L. Cohen Dept. of Mathematics Stanford University
The Topology of Fiber Bundles Lecture Notes Ralph L. Cohen Dept. of Mathematics Stanford University Contents Introduction v Chapter 1. Locally Trival Fibrations 1 1. Definitions and examples 1 1.1. Vector
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationSOME PROPERTIES OF SYMBOL ALGEBRAS OF DEGREE THREE
SOME PROPERTIES OF SYMBOL ALGEBRAS OF DEGREE THREE CRISTINA FLAUT and DIANA SAVIN Communicated by the former editorial board In this paper, we study some properties of the matrix representations of the
More informationarxiv: v2 [math.ag] 10 Jul 2011
DEGREE FORMULA FOR THE EULER CHARACTERISTIC OLIVIER HAUTION arxiv:1103.4076v2 [math.ag] 10 Jul 2011 Abstract. We give a proof of the degree formula for the Euler characteristic previously obtained by Kirill
More informationGENERATING SETS KEITH CONRAD
GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors
More informationSHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
More informationBreaking The Code. Ryan Lowe. Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and
Breaking The Code Ryan Lowe Ryan Lowe is currently a Ball State senior with a double major in Computer Science and Mathematics and a minor in Applied Physics. As a sophomore, he took an independent study
More informationCHAPTER 1 BASIC TOPOLOGY
CHAPTER 1 BASIC TOPOLOGY Topology, sometimes referred to as the mathematics of continuity, or rubber sheet geometry, or the theory of abstract topological spaces, is all of these, but, above all, it is
More informationZERODIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS
ZERODIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We
More informationit is easy to see that α = a
21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UF. Therefore
More information