On signatures associated with ramified coverings and embedding problems

J. Wood; Emery Thomas

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Displaying similar documents to “On signatures associated with ramified coverings and embedding problems”

On the Extension of Certain Maps with Values in Spheres

Carlos Biasi, Alice K. M. Libardi, Pedro L. Q. Pergher, Stanisław Spież (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m-2)-dimensional submanifold which is homologous to zero in E. Let $S^{n - 2} \subset S ⁿ$ be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if $h : V \to S^{n - 2}$ is a smooth map, then h extends to a smooth map g: E → Sⁿ transverse to $S^{n - 2}$ and with $g^{- 1} (S^{n - 2}) = V$ . Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental...

Commuting involutions whose fixed point set consists of two special components

Pedro L. Q. Pergher, Rogério de Oliveira (2008)

Fundamenta Mathematicae

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Let Fⁿ be a connected, smooth and closed n-dimensional manifold. We call Fⁿ a manifold with property when it has the following property: if $N^{m}$ is any smooth closed m-dimensional manifold with m > n and $T : N^{m} \to N^{m}$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. Examples of manifolds with this property are: the real, complex and quaternionic even-dimensional projective spaces $R P^{2 n}$ , $C P^{2 n}$ and $H P^{2 n}$ , and the connected sum of $R P^{2 n}$ and any number of copies of Sⁿ × Sⁿ, where Sⁿ is the n-sphere...

$f$ -biminimal maps between Riemannian manifolds

Yan Zhao, Ximin Liu (2019)

Czechoslovak Mathematical Journal

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We give the definition of $f$ -biminimal submanifolds and derive the equation for $f$ -biminimal submanifolds. As an application, we give some examples of $f$ -biminimal manifolds. Finally, we consider $f$ -minimal hypersurfaces in the product space $ℝ^{n} \times 𝕊^{1} (a)$ and derive two rigidity theorems.

A genericity theorem for algebraic stacks and essential dimension of hypersurfaces

Zinovy Reichstein, Angelo Vistoli (2013)

Journal of the European Mathematical Society

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We compute the essential dimension of the functors Forms $_{n, d}$ and Hypersurf $_{n, d}$ of equivalence classes of homogeneous polynomials in $n$ variables and hypersurfaces in $ℙ^{n - 1}$ , respectively, over any base field $k$ of characteristic $0$ . Here two polynomials (or hypersurfaces) over $K$ are considered equivalent if they are related by a linear change of coordinates with coefficients in $K$ . Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application...

$Z ₂^{k}$ -actions with a special fixed point set

Pedro L. Q. Pergher, Rogério de Oliveira (2005)

Fundamenta Mathematicae

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Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^{m}$ is any smooth and closed m-dimensional manifold with m > n and $T : N^{m} \to N^{m}$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions $(M^{m}; Φ)$ of the group $G = Z ₂^{k}$ on closed smooth m-dimensional manifolds $M^{m}$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F....

Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Frédéric Campana, Thomas Peternell (2011)

Bulletin de la Société Mathématique de France

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We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold $X$ have a pseudo-effective (instead of generically nef) determinant. A first consequence is that $X$ is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of $X$ is not covered by compact positive-dimensional analytic...

On the principle of real moduli flexibility: perfect parametrizations

Edoardo Ballico, Riccardo Ghiloni (2014)

Annales Polonici Mathematici

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Let V be a real algebraic manifold of positive dimension. The aim of this paper is to show that, for every integer b (arbitrarily large), there exists a trivial Nash family $= {V_{y}}_{y \in R^{b}}$ of real algebraic manifolds such that V₀ = V, is an algebraic family of real algebraic manifolds over $y \in R^{b} ∖ 0$ (possibly singular over y = 0) and is perfectly parametrized by $R^{b}$ in the sense that $V_{y}$ is birationally nonisomorphic to $V_{z}$ for every $y, z \in R^{b}$ with y ≠ z. A similar result continues to hold if V is a singular real algebraic...

A criterion for pure unrectifiability of sets (via universal vector bundle)

Silvano Delladio (2011)

Annales Polonici Mathematici

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Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let $π_{V}$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an $^{m}$ -measurable subset of ℝⁿ with $^{m} (A) < \infty$ . Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V, v) | V \in G (n, m), v \in V$ such that, for all P ∈ A, one has $^{m (n - m)} (V \in G (n, m) | (V, π_{V} (P)) \in Z) > 0$ . One can replace “for all P ∈ A” by “for...

Bounds for the characteristic rank and cup-length of oriented Grassmann manifolds

Tomáš Rusin (2018)

Archivum Mathematicum

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We estimate the characteristic rank of the canonical $k$ –plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n, k}$ . We then use it to compute uniform upper bounds for the $ℤ_{2}$ –cup-length of ${\tilde{G}}_{n, k}$ for $n$ belonging to certain intervals.

The canonical constructions of connections on total spaces of fibred manifolds

Włodzimierz M. Mikulski (2024)

Archivum Mathematicum

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We classify classical linear connections $A (Γ, Λ, Θ)$ on the total space $Y$ of a fibred manifold $Y \to M$ induced in a natural way by the following three objects: a general connection $Γ$ in $Y \to M$ , a classical linear connection $Λ$ on $M$ and a linear connection $Θ$ in the vertical bundle $V Y \to Y$ . The main result says that if $\dim (M) \geq 3$ and $\dim (Y) - \dim (M) \geq 3$ then the natural operators $A$ under consideration form the $17$ dimensional affine space.

A characterization of a certain real hypersurface of type $(A_{2})$ in a complex projective space

Byung Hak Kim, In-Bae Kim, Sadahiro Maeda (2017)

Czechoslovak Mathematical Journal

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In the class of real hypersurfaces $M^{2 n - 1}$ isometrically immersed into a nonflat complex space form ${\tilde{M}}_{n} (c)$ of constant holomorphic sectional curvature $c$ $(\neq 0)$ which is either a complex projective space $ℂ P^{n} (c)$ or a complex hyperbolic space $ℂ H^{n} (c)$ according as $c > 0$ or $c < 0$ , there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds....

On the Helmholtz operator of variational calculus in fibered-fibered manifolds

W. M. Mikulski (2007)

Annales Polonici Mathematici

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A fibered-fibered manifold is a surjective fibered submersion π: Y → X between fibered manifolds. For natural numbers s ≥ r ≤ q an (r,s,q)th order Lagrangian on a fibered-fibered manifold π: Y → X is a base-preserving morphism $λ : J^{r, s, q} Y \to ⋀^{d i m X} T * X$ . For p= max(q,s) there exists a canonical Euler morphism $(λ) : J^{r + s, 2 s, r + p} Y \to * Y \otimes ⋀^{d i m X} T * X$ satisfying a decomposition property similar to the one in the fibered manifold case, and the critical fibered sections σ of Y are exactly the solutions of the Euler-Lagrange equation $(λ) \circ j^{r + s, 2 s, r + p} σ = 0$ . In the present...

The signature package on Witt spaces

Pierre Albin, Éric Leichtnam, Rafe Mazzeo, Paolo Piazza (2012)

Annales scientifiques de l'École Normale Supérieure

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In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold $X$ which satisfies the Witt condition. This construction, which is inductive over the ‘depth’ of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index—the analytic signature...

Lagrangians and Euler morphisms on fibered-fibered frame bundles from projectable-projectable classical linear connections

Anna Bednarska (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We classify all $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $A$ transforming projectable-projectable torsion-free classical linear connections $\nabla$ on fibered-fibered manifolds $Y$ of dimension $(m_{1}, m_{2}, n_{1}, n_{2})$ into $r$ th order Lagrangians $A (r)$ on the fibered-fibered linear frame bundle $L^{f i b - f i b} (Y)$ on $Y$ . Moreover, we classify all $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $B$ transforming projectable-projectable torsion-free classical linear connections r on fiberedfibered manifolds $Y$ of dimension $(m_{1}, m_{2}, n_{1}, n_{2})$ into Euler morphism $B (\nabla)$ on $L^{f i b - f i b} (Y)$ . These classifications can be expanded on...

Universal acyclic resolutions for arbitrary coefficient groups

Michael Levin (2003)

Fundamenta Mathematicae

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We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dimension ≤ n+1 and a surjective $U V^{n - 1}$ -map r: Z → X such that for every abelian group G and every integer k ≥ 2 such that $d i m_{G} X \leq k \leq n$ we have $d i m_{G} Z \leq k$ and r is G-acyclic.

A note on the cohomology ring of the oriented Grassmann manifolds ${\tilde{G}}_{n, 4}$

Tomáš Rusin (2019)

Archivum Mathematicum

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We use known results on the characteristic rank of the canonical $4$ –plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n, 4}$ to compute the generators of the $ℤ_{2}$ –cohomology groups $H^{j} ({\tilde{G}}_{n, 4})$ for $n = 8, 9, 10, 11$ . Drawing from the similarities of these examples with the general description of the cohomology rings of ${\tilde{G}}_{n, 3}$ we conjecture some predictions.

Cycles on algebraic models of smooth manifolds

Wojciech Kucharz (2009)

Journal of the European Mathematical Society

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Every compact smooth manifold $M$ is diffeomorphic to a nonsingular real algebraic set, called an algebraic model of $M$ . We study modulo 2 homology classes represented by algebraic subsets of $X$ , as $X$ runs through the class of all algebraic models of $M$ . Our main result concerns the case where $M$ is a spin manifold.

On lifting of connections to Weil bundles

Jan Kurek, Włodzimierz M. Mikulski (2012)

Annales Polonici Mathematici

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We prove that the problem of finding all $ℳ f_{m}$ -natural operators $B : Q ⟿ Q T^{A}$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_{M} (\nabla)$ on the Weil bundle $T^{A} M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳ f_{m}$ -natural operators $C : Q ⟿ (T ¹_{p - 1}, T * \otimes T * \otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_{M} (\nabla) : T ¹_{p - 1} M = ⨁_{M}^{p - 1} T M \to T * M \otimes T * M \otimes T M$ .

On dimensionally restricted maps

H. Murat Tuncali, Vesko Valov (2002)

Fundamenta Mathematicae

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Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an $F_{σ}$ -subset $A_{k}$ of X such that $d i m A_{k} \leq k$ and the restriction $f | (X ∖ A_{k})$ is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...

Canonical contact forms on spherical CR manifolds

Wei Wang (2003)

Journal of the European Mathematical Society

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We construct the CR invariant canonical contact form $can (J)$ on scalar positive spherical CR manifold $(M, J)$ , which is the CR analogue of canonical metric on locally conformally flat manifold constructed by Habermann and Jost. We also construct another canonical contact form on the Kleinian manifold $Ω (Γ) / Γ$ , where $Γ$ is a convex cocompact subgroup of ${Aut}_{C R} S^{2 n + 1} = P U (n + 1, 1)$ and $Ω (Γ)$ is the discontinuity domain of $Γ$ . This contact form can be used to prove that $Ω (Γ) / Γ$ is scalar positive (respectively, scalar negative, or scalar vanishing)...

Liftings of 1-forms to $(J^{r} T ) $

Włodzimierz M. Mikulski (2002)

Colloquium Mathematicae

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Let $J^{r} T * M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r} T * M) *$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r} T * M) *$ is given.

The generic dimension of the first derived system

Robert P. Buemi (1978)

Annales de l'institut Fourier

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Any $r$ -dimensional subbundle of the cotangent bundle on an $n$ -dimensional manifold $M$ partitions $M$ into subsets $M_{0}, ..., M_{m}$ ( $m$ being the minimum of $r$ and $C (n - r, 2)$ , the combinations of $n - r$ things taken 2 at a time). $M_{i}$ is the set on which the first derived systems of the subbundle has codimension $i$ . In this paper we prove the following: Theorem. Let $s \geq 2$ and let $Q$ be a generic $C^{s}$ $r$ -dimensional subbundle of the cotangent bundle of an $n$ -dimensional manifold $M$ . The codimension...