On the pontryagin-steenrod-wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π ⁿ (M) →H ⁿ (M; ℤ) by the formula degf =f*[S ⁿ ], where [S ⁿ ] εH ⁿ (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH ₂(M, ℤ/2ℤ) such thatβ ·w ₂(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ ₂ α ·w ₂(M)=0, whereρ ₂: ℤ → ℤ/2ℤ is reduction modulo 2.     

Classification of framed links in 3-manifolds

We present a short and complete proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in detail. Let M be a connected oriented closed smooth 3-manifold, L ₁(M) be the set of framed links in M up to a framed cobordism, and deg: L ₁(M) → H ₁(M; ℤ) be the map taking a framed link to its homology class. Then for each α ∈ H ₁(M; ℤ) there is a one-to-one correspondence between the set deg⁻¹ α and the group ℤ_2d(α), where d(α) is the divisibility of the projection of α to the free part of H ₁(M; ℤ).     

On nonnegatively curved 4-manifolds with discrete symmetry

Let M be the closed, simply connected, 4-manifold with nonnegative sectional curvature, called a nonnegatively curved 4-manifold, with an effective and isometric Z _m -action for a positive integer m ≧ 61⁷. Assume that Z _m acts trivially on the homology of M. The goal of this short paper is to prove that if the fixed point set of any nontrivial element of Z _m has at most one two-dimensional component, then M is homeomorphic to S ⁴, # _i ^l =1S ² × S ², l = 1, 2, or # _j ^k = 1 ± CP ², k = 1, 2, 3, 4, 5. The main strategy of this paper is to give an upper bound of the Euler characteristic χ(M) under the homological assumption of a Z _m -action as above by using the Lefschetz fixed point formula.     

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C ₀(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak^*-continuous. Given a discrete semigroup S, the convolution algebra ℓ ¹(S) also carries a coproduct. In this paper we examine preduals for ℓ ¹(S) making both the product and the coproduct weak^*-continuous. Under certain conditions on S, we show that ℓ ¹(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ ¹(S) when S is either ℤ₊×ℤ or (ℕ,⋅).     

The relative trace formula for groups with involution

A theorem of Bourgain states that the harmonic measure for a domain in ℝ ^d is supported on a set of Hausdorff dimension strictly less thand [2]. We apply Bourgain’s method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of ℤ ^d ,d≥2. By refining the argument, we prove that for allβ>0 there existsρ(d,β)<d andN(d,β), such that for anyn>N(d,β), anyx ∈ ℤ ^d , and anyA ⊂ {1,…,n} ^d •{y∈ℤ whereν _A,x (y) denotes the probability thaty is the first entrance point of the simple random walk starting atx intoA. Furthermore,ρ must converge tod asβ → ∞. Supported by Swiss NF grant 20-55648.98.     

On the singular set of stationary harmonic maps

LetM andN be compact riemannian manifolds, andu a stationary harmonic map fromM toN. We prove thatH ⁿ⁻² (Σ)=0, wheren=dimM and Σ is the singular set ofu. This is a generalization of a result of C. Evans [7], where this is proved in the special caseN is a sphere. We also prove that, ifu is a weakly harmonic map inW ^1,n (M, N), thenu is smooth. This extends results of F. Hélein for the casen=2, or the caseN is a sphere ([9], [10]).     

The density of rational points on non-singular hypersurfaces

LetF(x) =F[x₁,…,x_n]∈ℤ[x₁,…,x_n] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ ⁿ ;F(x)=0, |x|⩽X}, where . It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X ^n−2+2/n for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields. However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process. Dedicated to the memory of Professor K G Ramanathan     

An estimate of the pinching constant of minimal hypersurfaces with constant scalar curvature in the unit sphere

LetM ⁿ (n>3) be a closed minimal hypersurface with constant scalar curvature in the unit sphereS ⁿ⁺¹ (1) andS the square of the length of its second fundamental form. In this paper we prove thatS>n implies estimates of the formS>n+cn−d withc≥1/4. For example, forn>17 andS>n we proveS>n+1/4n which is sharper than a recent result of the authors [5] The second author's research was supported by NNSFC, FECC and CPSF.     

Distance and fractional isomorphism in Steiner triple systems

In [8], Quattrochi and Rinaldi introduced the idea ofn ⁻¹-isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. The main result is that for any positive integerN, there existsv ₀(N) such that for all admissiblev≥v ₀(N) and for each STS(v) (sayS), there exists an STS(v) (sayS′) such that for somen>N, S is strictlyn ⁻¹-isomorphic toS′. We also prove that for all admissiblev≥13, there exist two STS(v)s which are strictly 2⁻¹-isomorphic. Define the distance between two Steiner triple systemsS andS′ of the same order to be the minimum volume of a tradeT which transformsS into a system isomorphic toS′. We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly 2⁻¹-isomorphic and 3⁻¹-isomorphic pairs of STS(15)s.     

A sharper tits alternative for 3-manifold groups

The following theorem is proven. LetM be a closed, orientable, irreducible 3-manifold such that rankH ₁(M, ℤ/pℤ)≥3 for some primep. Then either π₁(M) is virtually solvable or it contains a free group of rank 2.     

On polynomial symbols for subdivision schemes

Given a dilation matrix A :ℤ^d→ℤ^d, and G a complete set of coset representatives of 2π(A ^−Tℤ^d/ℤ^d), we consider polynomial solutions M to the equation ∑ _g∈G M(ξ+g)=1 with the constraints that M≥0 and M(0)=1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying Strang–Fix conditions of arbitrary order. Research partially supported by the Danish Technical Science Foundation, Grant No. 9701481, and by the Danish SNF-PDE network.     

Projections in the space<Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> and the corona theorem for subdomains of coverings of finite bordered Riemann surfaces

LetM be a non-compact connected Riemann surface of a finite type, andR⋐M be a relatively compact domain such thatH ₁(M,Z)=H ₁(R,Z). Let be a covering. We study the algebraH ^∞(U) of bounded holomorphic functions defined in certain subdomains . Our main result is a Forelli type theorem on projections inH ^∞(D). Research supported in part by NSERC.     

Differentiable structures with zero entropy on simply connected 4-manifolds

We show that a closed 4-dimensional simply connected topological manifoldM admits a differentiable structure with aC Riemannian metric whose geodesic flow has zero topological entropy if and only ifM is homeomorphic toS ⁴, ²,S ²×S ², or ²#².     

Limits of functions and elliptic operators

We show that a subspaceS of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are thatS is closed inL ² (M) and that if a sequence of functions fn in ƒ_n converges inL ²(M), then so do the partial derivatives of the functions ƒ_n.     

Deformations of circle-valued Morse functions on surfaces

Let M be a smooth connected orientable compact surface and let F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) be a space of all Morse functions f : M → S ¹ without critical points on ∂M such that, for any connected component V of ∂M, the restriction f : V → S ¹ is either a constant map or a covering map. The space F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) is endowed with the C ^∞-topology. We present the classification of connected components of the space F_cov ( M,S¹ ) {\mathcal{F}_{{\rm cov} }}\left( {M,{S^1}} \right) . This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on ∂M.     

Minimal entropy and collapsing with curvature bounded from below

We show that if a closed manifold M admits an ℱ-structure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a non-trivial S ¹-action. As a corollary we obtain that the simplicial volume of a manifold admitting an ℱ-structure is zero.?We also show that if M admits an ℱ-structure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is non-negative.?We show that ℱ-structures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5-manifold.?We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S ⁴, ℂP ², ²,S ²×S ²and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁴,ℂP ²,S ²×S ²,ℂP ²#  ² or ℂP ²# ℂP ². Finally, suppose that M is a closed simply connected 5-manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S ⁵,S ³×S ², then on trivial S ³-bundle over S ² or the Wu-manifold SU(3)/SO(3). Oblatum 13-III-2002 & 12-VIII-2002?Published online: 8 November 2002 G.P. Paternain was partially supported by CIMAT, Guanajuato, México.?J. Petean is supported by grant 37558-E of CONACYT.     

A geometrical problem arising in a signal restoration algorithm

Let ℤ_2N={0, ..., 2N-1} denote the group of integers modulo 2N, and let L be the space of all real functions of ℤ_2N which are supported on {0,...N−1}. The spectral phase of a function f:ℤ_2N→ℝ is given by φ_f(k)=arg for k ∈ ℤ_2N, where denotes the discrete Fourier transforms of f. For a fixed s∈L let K_s denote the cone of all f:ℤ_2N→ℝ which satisfy φ_f ≡ φ_s and let M_s be its linear span. The angle α_s between M_s and L determines the convergence rate of the signal restoration from phase algorithm of Levi and Stark [3]. Here we prove the following conjectures of Urieli et al. [7] who verified them for the N≤3 case:

Sur le 2-groupe de classes d'idéaux de ℚ(√d,i)

Let ℕ,i=√−1,k=ℚ(√d,i) andC ₂ the 2-part of the class group ofk. Our goal is to determine alld such thatC ₂⋍ℤ/2ℤ×ℤ/2ℤ. Soientd∈ℕ,i=√−1,k=ℚ(√d,i), etC ₂ la 2-partie du groupe de classes dek. On s'intéresse à déterminer tous lesd tel queC ₂⋍ℤ/2ℤ×ℤ/2ℤ.      

Compactifying coverings of 3-manifolds

If a finitely presented groupG is negatively curved, automatic or asynchronously automatic thenG has an asynchronously bounded “almost prefix closed” combing. Results in [Br₁] and [E] imply that the fundamental group of any closed 3-manifold satisfying Thurston's geometrization conjecture has an asynchronously bounded, almost prefix closed combing. MAIN THEOREM. IfM is a compactP ²-irreducible 3-manifold,π ₁ (M) has an asynchronously bounded, almost prefix closed combing, andH, a subgroup ofπ ₁ (M), is quasiconvex with respect to this combing, then the cover ofM corresponding toH is a missing boundary manifold.     

Convergence of the median filterings of sequences

Suppose thatx=|x(n)|_n∈ℤ is a sequence of real numbers. For eachp∈ℕ,x ^p =|x ^p (n)|_n∈ℤis the resulting sequence ofx throughp times median filterings with window 2k+1. It is proved that whenp→∞, bothx ^(2p) andx(² p}-1) are convergent. Thus the problem of convergence of the median filters of infinite-length sequences is completely solved. Project supported by the National Natural Science Foundation of China (Grant No. 16971047).