Mapping Degree for A-Proper Operators

For the single-valued case the notion of degree has been given recent expression by papers of Dold [5] for the finite dimensional case, and by Leray-Schauder [8] for the locally convex linear topological space. Klee [7] has removed this restriction by use of shrinkable in place of convex neighborhoods with the central role filled by a form of (2.15) below. For set-valued maps a modern formulation is, for instance, to be found in Gorniewicz-Granas [6]. These contributions relate the degree to the Lefschetz number, and the set-valued maps are required to map points into acyclic sets; that is to say, into "swollen points".

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Mapping degree and Euler characteristic

Kodai Mathematical Journal ◽

10.2996/kmj/1143122391 ◽

2006 ◽

Vol 29 (1) ◽

pp. 144-162 ◽

Cited By ~ 1

Author(s):

T. Fukui ◽

A. Khovanskii

Keyword(s):

Euler Characteristic ◽

Mapping Degree

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ON INTEGERS OCCURRING AS THE MAPPING DEGREE BETWEEN QUASITORIC 4-MANIFOLDS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788716000598 ◽

2016 ◽

Vol 103 (3) ◽

pp. 289-312

Author(s):

ÐORÐE B. BARALIĆ

Keyword(s):

Number Theory ◽

Quadratic Forms ◽

Point Of View ◽

Theoretical Point ◽

Mapping Degree ◽

Mapping Degrees

We study the set$D(M,N)$of all possible mapping degrees from$M$to$N$when$M$and$N$are quasitoric$4$-manifolds. In some of the cases, we completely describe this set. Our results rely on Theorems proved by Duan and Wang and the sets of integers obtained are interesting from the number theoretical point of view, for example those representable as the sum of two squares$D(\mathbb{C}P^{2}\sharp \mathbb{C}P^{2},\mathbb{C}P^{2})$or the sum of three squares$D(\mathbb{C}P^{2}\sharp \mathbb{C}P^{2}\sharp \mathbb{C}P^{2},\mathbb{C}P^{2})$. In addition to the general results about the mapping degrees between quasitoric 4-manifolds, we establish connections between Duan and Wang’s approach, quadratic forms, number theory and lattices.

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