Realization of numbers as the degrees of maps between manifolds
type: Item
instance of: scholarly article
title: Realization of numbers as the degrees of maps between manifolds
zbMATH Open document ID: 1205.55003
author: Young Min Lee
author: Fei Xu
publication date: 2010-11-17T00:00:00Z
review text: Let \(M\) and \(N\) be oriented closed manifolds of the same dimension and call \(D(M,N)\) the set of \(\text{deg}(f)\) where \(f\) ranges over all continuous maps \(f:M\to N\). The authors conjecture that either \(D(M,N)=\{0\}\) or \(D(M,N)\) contains infinitely many integers. This conjecture is equivalent to the conjecture that \(D(M,M)\) contains an element different from \(\pm1\) for any closed oriented manifold. The authors consider the case where \(M\) belongs to the set \(V(n)\) of closed \((n-1)\)-connected \(2n\)-manifolds. The elements of \(V(n)\) are classified by the cup product \(H^n(M)\times H^n(M)\to H^{2n}(M)\) and a map \(H^n(M)\to\pi_{n-1}(\text{SO}(n))\) for \(n\geq3\). When \(n\) is even the cup product is an integral quadratic form, and \(V(4n)\) is parameterized by \(\prod_{i\geq0}(S(i)\times\mathbb{Z}^i/\sim\), cf. [\textit{H. Duan} and \textit{S. Wang}, Math. Z. 244, 67--89 (2003; Zbl 1022.55003)]. Here \(S(i)\) denotes the set of all symmetric \((i,i)\) matrices with integer entries and determinant \(\pm1\), and \(\sim\) is defined by \((A,a)\sim (A',a')\) iff there is a \(T\in\text{GL}(i,\mathbb{Z})\) such that \(A'=T^\top AT\) and \(A'=aT\). Let then \(M,N\in V(4n)\) be represented by \((A,a)\in S(i)\times\mathbb{Z}^i\) and \(B(b)\times\mathbb{Z}^j\). Duan and Wang [loc. cit.] proved that \(k\in D(M,N)\) if \(X^\top AX=kB\) plus some congruences relating \(k,a,a',b,b'\). The present authors explicitly determine all \(k\) satisfying the last condition.
Mathematics Subject Classification ID: 55M25
Mathematics Subject Classification ID: 11E12
zbMATH DE Number: 5816562
zbMATH Keywords: degree of a map
zbMATH Keywords: integral quadratic form
zbMATH Keywords: oriented closed manifold
reviewed by: Christian Fenske
MaRDI profile type: Publication
full work available at URL: s10114-010-9102-7
OpenAlex ID: W2004122163
cites work: Some Historical Remarks Concerning Degree Theory
cites work: Classification of (n-1)-connected 2n-manifolds
cites work: Non-zero degree maps between \(2n\)-manifolds
cites work: The degrees of maps between manifolds
cites work: Computing degree of maps between manifolds
cites work: Brauer–Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms
cites work: The Integral Representations of Quadratic Forms Over Local Fields
cites work: Q5670687
DOI: 10.1007/S10114-010-9102-7
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5816562