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

##### MSC:
 55M25 Degree, winding number 11E12 Quadratic forms over global rings and fields
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##### References:
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