Realization of numbers as the degrees of maps between manifolds. (English) Zbl 1205.55003

Lee, Young Min; Xu, Fei

Acta Math. Sin., Engl. Ser. 26, No. 8, 1413-1424 (2010).

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. [H. Duan and 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.

Reviewer: Christian Fenske (Gießen)

MSC:

55M25	Degree, winding number
11E12	Quadratic forms over global rings and fields

Keywords:

degree of a map; integral quadratic form; oriented closed manifold

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