Immersions and Embeddings Up to Cobordism

1971 ◽  
Vol 23 (6) ◽  
pp. 1102-1115 ◽  
Author(s):  
Richard L. W. Brown
Keyword(s):  
Projective Space ◽  
Compact Manifold ◽  
Real Projective Space ◽  
Binary Expansion ◽  
Simple Argument ◽  
Prove Theorem ◽  
Definition Of

In 1944 Whitney proved that any differentiable n-manifold (n ≧ 2) can be (differentiably) immersed in R2n–1[15] and embedded in R2n [14]. Whitney's results are best possible when n = 2r. (One uses a simple argument involving the dual Stiefel-Whitney classes of real projective space Pn. See [9, pp. 14, 15].) However, there is a widely known conjecture that any R-manifold (n ≧ 2) immerses in R2n–α(n) and embeds in R2n–α(n)+1. Here, α(n) denotes the number of ones in the binary expansion of n. We prove (Theorem 5.1) that every compact manifold is cobordant to a manifold that immerses in (2n – α(n))-space and embeds in (2n – α(n) + 1)-space. (See § 1 for the definition of cobordant manifolds.) It is well known that if the conjecture were true it would be the best possible result.

Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension

2018 ◽  
Vol 30 (2) ◽  
pp. 397-417 ◽  
Author(s):  
Natalia Cadavid-Aguilar ◽  
Jesús González ◽  
Darwin Gutiérrez ◽  
Aldo Guzmán-Sáenz ◽  
Adriana Lara
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Motion Planning ◽  
Product Space ◽  
Projective Spaces ◽  
Topological Complexity ◽  
Real Projective Space ◽  
Binary Expansion ◽  
The Real

AbstractThes-th higher topological complexity{\operatorname{TC}_{s}(X)}of a spaceXcan be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when{X=\operatorname{\mathbb{R}P}^{m}}, the real projective space of dimensionm. In particular, we describe a number{r(m)}, which depends on the structure of zeros and ones in the binary expansion ofm, and with the property that{0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)}for{s\geq r(m)}, where{\delta_{s}(m)=(0,1,0)}for{m\equiv(0,1,2)\bmod 4}. Such an estimation for{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}appears to be closely related to the determination of the Euclidean immersion dimension of{\operatorname{\mathbb{R}P}^{m}}. We illustrate the phenomenon in the case{m=3\cdot 2^{a}}. In addition, we show that, for large enoughsand evenm,{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}is characterized as the smallest positive integer{t=t(m,s)}for which there is a suitable equivariant map from Davis’ projective product space{\mathrm{P}_{\mathbf{m}_{s}}}to the{(t+1)}-st join-power{((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}}. This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating{\operatorname{TC}_{2}}to the immersion dimension of real projective spaces.

scholarly journals A wall-crossing formula for degrees of Real central projections

2014 ◽  
Vol 25 (04) ◽  
pp. 1450038 ◽  
Author(s):  
Christian Okonek ◽  
Andrei Teleman
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Pole Placement ◽  
Real Projective Space ◽  
Central Projections ◽  
Wall Crossing ◽  
Subspace Problem ◽  
Wall Crossing Formula ◽  
Real Subspace ◽  
Crucial Assumption

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

scholarly journals On compactness of isospectral conformal, metrics of 4-manifolds

1995 ◽  
Vol 140 ◽  
pp. 77-99 ◽  
Author(s):  
Xingwang Xu
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Compact Manifold ◽  
Riemannian Metrics ◽  
Conformal Class ◽  
Conformal Metrics ◽  
Laplace Operators ◽  
Definition Of ◽  
The Laplace Operator ◽  
Isometry Class

In this paper, we are interested in the compactness of isospectral conformal metrics in dimension 4.Let us recall the definition of the isospectral metrics. Two Riemannian metrics g, g′ on a compact manifold are said to be isospectral if their associated Laplace operators on functions have identical spectrum. There are now numeruos examples of compact Riemannian manifolds which admit more than two metrics such that they are isospectral but not isometric. That is to say that the eigenvalues of the Laplace operator Δ on the functions do not necessarily determine the isometry class of (M, g). If we further require the metrics stay in the same conformal class, the spectrum of Laplace operator still does not determine the metric uniquely ([BG], [BPY]).

About one definition of the canonical plane of a surface in a projective space

1992 ◽  
Vol 59 (2) ◽  
pp. 733-737
Author(s):  
N. I. Kovantsov ◽  
P. A. Tadeev
Keyword(s):  
Projective Space ◽  
Definition Of
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