The semi-index product formula

Jerzy Jezierski

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A Lefschetz-type coincidence theorem

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A Lefschetz-type coincidence theorem for two maps f,g: X → Y from an arbitrary topological space to a manifold is given: $I_{f g} = λ_{f g}$ , that is, the coincidence index is equal to the Lefschetz number. It follows that if $λ_{f g} \neq 0$ then there is an x ∈ X such that f(x) = g(x). In particular, the theorem contains well-known coincidence results for (i) X,Y manifolds, f boundary-preserving, and (ii) Y Euclidean, f with acyclic fibres. It also implies certain fixed point results for multivalued maps with “point-like”...

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For the Brown-Peterson spectrum BP at the prime 3, $v_{2}$ denotes Hazewinkel’s second polynomial generator of $B P_{*}$ . Let $L_{2}$ denote the Bousfield localization functor with respect to $v_{2}^{- 1} B P$ . A typical example of type one finite spectra is the mod 3 Moore spectrum M. In this paper, we determine the homotopy groups $π_{*} (L_{2} M \land X)$ for the 8 skeleton X of BP.

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It is shown that for n ≥ 2 and p > 2, where p is not an even integer, the only balls in the Carathéodory distance on $H_{p, n} = z \in ℂ^{n} : ∥ z ∥_{p} < 1$ which are balls with respect to the complex $l_{p}$ norm in $ℂ^{n}$ are those centered at the origin.