The Wecken property of the projective plane

Boju Jiang

Displaying similar documents to “The Wecken property of the projective plane”

On the generalized Massey–Rolfsen invariant for link maps

A. Skopenkov (2000)

Fundamenta Mathematicae

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For $K = K_{1} ⊔ . . . ⊔ K_{s}$ and a link map $f : K \to ℝ^{m}$ let $K^{\sim} = ⊔_{i < j} K_{i} \times K_{j}$ , define a map $f^{\sim} : K^{\sim} \to S^{m - 1}$ by $f^{\sim} (x, y) = (f x - f y) / | f x - f y |$ and a (generalized) Massey-Rolfsen invariant $α (f) \in π^{m - 1} (K)$ to be the homotopy class of $f^{\sim}$ . We prove that for a polyhedron K of dimension ≤ m - 2 under certain (weakened metastable) dimension restrictions, α is an onto or a 1 - 1 map from the set of link maps $f : K \to ℝ^{m}$ up to link concordance to $π^{m - 1} (K^{\sim})$ . If $K_{1}, . . ., K_{s}$ are closed highly homologically connected manifolds of dimension $p_{1}, . . ., p_{s}$ (in particular, homology spheres), then $π^{m - 1} (K^{\sim}) ≅ \oplus_{i < j} π_{p_{i} + p_{j} - m + 1}^{S}$ .

Generalized Lefschetz numbers of pushout maps defined on non-connected spaces

Davide Ferrario (1999)

Banach Center Publications

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Let A, $X_{1}$ and $X_{2}$ be topological spaces and let $i_{1} : A \to X_{1}$ , $i_{2} : A \to X_{2}$ be continuous maps. For all self-maps $f_{A} : A \to A$ , $f_{1} : X_{1} \to X_{1}$ and $f_{2} : X_{2} \to X_{2}$ such that $f_{1} i_{1} = i_{1} f_{A}$ and $f_{2} i_{2} = i_{2} f_{A}$ there exists a pushout map f defined on the pushout space $X_{1} ⊔_{A} X_{2}$ . In [F] we proved a formula relating the generalized Lefschetz numbers of f, $f_{A}$ , $f_{1}$ and $f_{2}$ . We had to assume all the spaces involved were connected because in the original definition of the generalized Lefschetz number given by Husseini in [H] the space was assumed to be connected. So, to extend the result of [F]...

The Nielsen coincidence theory on topological manifolds

Jerzy Jezierski (1993)

Fundamenta Mathematicae

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We generalize the coincidence semi-index introduced in [D-J] to pairs of maps between topological manifolds. This permits extending the Nielsen theory to this class of maps.

Homotopy and homology groups of the n-dimensional Hawaiian earring

Katsuya Eda, Kazuhiro Kawamura (2000)

Fundamenta Mathematicae

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For the n-dimensional Hawaiian earring $ℍ_{n},$ n ≥ 2, $π_{n} (ℍ_{n}, o) ≃ ℤ^{ω}$ and $π_{i} (ℍ_{n}, o)$ is trivial for each 1 ≤ i ≤ n - 1. Let CX be the cone over a space X and CX ∨ CY be the one-point union with two points of the base spaces X and Y being identified to a point. Then $H_{n} (X \lor Y) ≃ H_{n} (X) \oplus H_{n} (Y) \oplus H_{n} (C X \lor C Y)$ for n ≥ 1.

On the computation of the Nielsen numbers and the converse of the Lefschetz coincidence theorem

Peter Wong (1992)

Fundamenta Mathematicae

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Let $f, g : M_{1} \to M_{2}$ be maps where $M_{1}$ and $M_{2}$ are connected triangulable oriented n-manifolds so that the set of coincidences $C_{f, g} = x \in M_{1} | f (x) = g (x)$ is compact in $M_{1}$ . We define a Nielsen equivalence relation on $C_{f, g}$ and assign the coincidence index to each Nielsen coincidence class. In this note, we show that, for n ≥ 3, if $M_{2} = {\tilde{M}}_{2} / K$ where ${\tilde{M}}_{2}$ is a connected simply connected topological group and K is a discrete subgroup then all the Nielsen coincidence classes of f and g have the same coincidence index. In particular, when $M_{1}$ and $M_{2}$ are...

The homotopy type of the space of degree 0 immersed plane curves.

Hiroki Kodama, Peter W. Michor (2006)

Revista Matemática Complutense

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The space B = Imm (S, R) / Diff (S) of all immersions of rotation degree 0 in the plane modulo reparameterizations has homotopy groups π(B ) = Z, π(B ) = Z, and π(B ) = 0 for k ≥ 3.

Nielsen theory of transversal fixed point sets (with an appendix: $C^{\infty}$ and C0 fixed point sets are the same, by R. E. Greene)

Helga Schirmer (1992)

Fundamenta Mathematicae

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Examples exist of smooth maps on the boundary of a smooth manifold M which allow continuous extensions over M without fixed points but no such smooth extensions. Such maps are studied here in more detail. They have a minimal fixed point set when all transversally fixed maps in their homotopy class are considered. Therefore we introduce a Nielsen fixed point theory for transversally fixed maps on smooth manifolds without or with boundary, and use it to calculate the minimum number of...

The coincidence Nielsen number for maps into real projective spaces

Jerzy Jezierski (1992)

Fundamenta Mathematicae

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We give an algorithm to compute the coincidence Nielsen number N(f,g), introduced in [DJ], for pairs of maps into real projective spaces.

On holomorphically projective mappings from equiaffine generally recurrent spaces onto Kählerian spaces

Raad J. K. al Lami, Marie Škodová, Josef Mikeš (2006)

Archivum Mathematicum

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In this paper we consider holomorphically projective mappings from the special generally recurrent equiaffine spaces $A_{n}$ onto (pseudo-) Kählerian spaces ${\bar{K}}_{n}$ . We proved that these spaces $A_{n}$ do not admit nontrivial holomorphically projective mappings onto ${\bar{K}}_{n}$ . These results are a generalization of results by T. Sakaguchi, J. Mikeš and V. V. Domashev, which were done for holomorphically projective mappings of symmetric, recurrent and semisymmetric Kählerian spaces.