• Title/Summary/Keyword: rational homotopy

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RATIONAL HOMOTOPY TYPE OF MAPPING SPACES BETWEEN COMPLEX PROJECTIVE SPACES AND THEIR EVALUATION SUBGROUPS

  • Gatsinzi, Jean-Baptiste
    • Communications of the Korean Mathematical Society
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    • v.37 no.1
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    • pp.259-267
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    • 2022
  • We use L models to compute the rational homotopy type of the mapping space of the component of the natural inclusion in,k : ℂPn ↪ ℂPn+k between complex projective spaces and show that it has the rational homotopy type of a product of odd dimensional spheres and a complex projective space. We also characterize the mapping aut1 ℂPn → map(ℂPn, ℂPn+k; in,k) and the resulting G-sequence.

ON THE RATIONAL COHOMOLOGY OF MAPPING SPACES AND THEIR REALIZATION PROBLEM

  • Abdelhadi Zaim
    • Communications of the Korean Mathematical Society
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    • v.38 no.4
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    • pp.1309-1320
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    • 2023
  • Let f : X → Y be a map between simply connected CW-complexes of finite type with X finite. In this paper, we prove that the rational cohomology of mapping spaces map(X, Y ; f) contains a polynomial algebra over a generator of degree N, where N = max{i, πi(Y)⊗ℚ ≠ 0} is an even number. Moreover, we are interested in determining the rational homotopy type of map(𝕊n, ℂPm; f) and we deduce its rational cohomology as a consequence. The paper ends with a brief discussion about the realization problem of mapping spaces.

A PROSET STRUCTURE INDUCED FROM HOMOTOPY CLASSES OF MAPS AND A CLASSIFICATION OF FIBRATIONS

  • Yamaguchi, Toshihiro;Yokura, Shoji
    • Communications of the Korean Mathematical Society
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    • v.34 no.3
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    • pp.991-1004
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    • 2019
  • Firstly we consider preorders (not necessarily partial orders) on a canonical quotient of the set of the homotopy classes of continuous maps between two spaces induced by a certain equivalence relation ${\sim}_{{\varepsilon}R}$. Secondly we apply it to a classification of orientable fibrations over Y with fibre X. In the classification theorem of J. Stasheff [22] and G. Allaud [3], they use the set $[Y,\;Baut_1X]$ of homotopy classes of continuous maps from Y to $Baut_1X$, which is the classifying space for fibrations with fibre X due to A. Dold and R. Lashof [11]. In this paper we give a classification of fibrations using a preordered set (abbr., proset) structure induced by $[Y,\;Baut_1X]_{{\varepsilon}R}:=[Y,\;Baut_1X]/{\sim}_{{\varepsilon}R}$.

Homotopical triviality of entire rational maps to even dimensional spheres

  • Suh, Dong-Youp
    • Communications of the Korean Mathematical Society
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    • v.11 no.3
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    • pp.807-814
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    • 1996
  • Let $G = Z_2$. Let X be any compact connected orientable nonsingular real algebraic variety of dim X = k = odd with the trivial G action, and let Y be the unit sphere $S^{2n-k}$ with the antipodal action of G. Then we prove that any G invariant entire rational map $f : x \times Y \to S^{2n}$ is G homotopically trivial. We apply this result to prove that any entire rational map $g : X \times RP^{2n-k} \to S^{2n}$ is homotopically trivial.

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REAL POLYHEDRAL PRODUCTS, MOORE'S CONJECTURE, AND SIMPLICIAL ACTIONS ON REAL TORIC SPACES

  • Kim, Jin Hong
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.4
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    • pp.1051-1063
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    • 2018
  • The real moment-angle complex (or, more generally, real polyhedral product) and its real toric space have recently attracted much attention in toric topology. The aim of this paper is to give two interesting remarks regarding real polyhedral products and real toric spaces. That is, we first show that Moore's conjecture holds to be true for certain real polyhedral products. In general, real polyhedral products show some drastic difference between the rational and torsion homotopy groups. Our result shows that at least in terms of the homotopy exponent at a prime this is not the case for real polyhedral products associated to a simplicial complex whose minimal missing faces are all k-simplices with $k{\geq}2$. Moreover, we also show a structural theorem for a finite group G acting simplicially on the real toric space. In other words, we show that G always contains an element of order 2, and so the order of G should be even.

FIBREWISE INFINITE SYMMETRIC PRODUCTS AND M-CATEGORY

  • Hans, Scheerer;Manfred, Stelzer
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.4
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    • pp.671-682
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    • 1999
  • Using a base-point free version of the infinite symmetric product we define a fibrewise infinite symmetric product for any fibration $E\;\longrightarrow\;B$. The construction works for any commutative ring R with unit and is denoted by $R_f(E)\;l\ongrightarrow\;B$. For any pointed space B let $G_I(B)\;\longrightarrow\;B$ be the i-th Ganea fibration. Defining $M_R-cat(B):= inf{i\midR_f(G_i(B))\longrihghtarrow\;B$ admits a section} we obtain an approximation to the Lusternik-Schnirelmann category of B which satisfies .g.a product formula. In particular, if B is a 1-connected rational space of finite rational type, then $M_Q$-cat(B) coincides with the well-known (purely algebraically defined) M-category of B which in fact is equal to cat (B) by a result of K.Hess. All the constructions more generally apply to the Ganea category of maps.

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WHEN IS THE CLASSIFYING SPACE FOR ELLIPTIC FIBRATIONS RANK ONE?

  • YAMAGUCHI TOSHIHIRO
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.521-525
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    • 2005
  • We give a necessary and sufficient condition of a rationally elliptic space X such that the Dold-Lashof classifying space Baut1X for fibrations with the fiber X is rank one. It is only when X has the rational homotopy type of a sphere or the total space of a spherical fibration over a product of spheres.

A REMARK ON FORMALITY

  • Lee, Doobeum
    • Korean Journal of Mathematics
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    • v.6 no.2
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    • pp.313-318
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    • 1998
  • In this paper we prove two independent theorems concerning formality of a nilmanifold and a differential graded algebra using the well-known theorem of Deligne-Griffiths-Morgan-Sullivan. We first give a rational homotopy theoretic proof to the statement that a nilmanifold is formal if and only if it is a torus. And then we study some conditions with which formality of one dga implies formality of the other in an extension of dga's.

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A NOTE ON DERIVATIONS OF A SULLIVAN MODEL

  • Kwashira, Rugare
    • Communications of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.279-286
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    • 2019
  • Complex Grassmann manifolds $G_{n,k}$ are a generalization of complex projective spaces and have many important features some of which are captured by the $Pl{\ddot{u}}cker$ embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$ where $N=\(^n_k\)$. The problem of existence of cross sections of fibrations can be studied using the Gottlieb group. In a more generalized context one can use the relative evaluation subgroup of a map to describe the cohomology of smooth fiber bundles with fiber the (complex) Grassmann manifold $G_{n,k}$. Our interest lies in making use of techniques of rational homotopy theory to address problems and questions involving applications of Gottlieb groups in general. In this paper, we construct the Sullivan minimal model of the (complex) Grassmann manifold $G_{n,k}$ for $2{\leq}k<n$, and we compute the rational evaluation subgroup of the embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$. We show that, for the Sullivan model ${\phi}:A{\rightarrow}B$, where A and B are the Sullivan minimal models of ${\mathbb{C}}P^{N-1}$ and $G_{n,k}$ respectively, the evaluation subgroup $G_n(A,B;{\phi})$ of ${\phi}$ is generated by a single element and the relative evaluation subgroup $G^{rel}_n(A,B;{\phi})$ is zero. The triviality of the relative evaluation subgroup has its application in studying fibrations with fibre the (complex) Grassmann manifold.