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Session 35. Topological fixed point theory and related topics
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The spaces of proper and local maps are not homotopy equivalent |
Piotr Nowak-Przygodzki, Gdańsk University of Technology, Poland
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This is a joint work with Piotr Bartłomiejczyk
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Let \(\mathcal{M}(X,Y)\) be the set of all continuous maps \(f\colon D_f\to Y\)
such that \(D_f\) is an open subset of \(X\). Let us define the set of local maps
\[\mathcal{F}(n,k)=\{f\in\mathcal{M}(\mathbb{R}^{n+k},\mathbb{R}^n)
\mid \text{\(f^{-1}(0)\) is compact} \}\]
and the set of proper maps
\[\mathcal{P}(n,k)=\{f\in\mathcal{M}(\mathbb{R}^{n+k},\mathbb{R}^n) \mid
\text{\(f^{-1}(K)\) is compact for any compact set \(K\)} \}.\]
In [1] we introduce the topology
on the set of local maps in more general
setting and prove that the inclusion
\(\mathcal{P}(n,k)\subset\mathcal{F}(n,k)\)
is a weak homotopy equivalence.
We denote by \(\mathcal{F}_0(n,k)\) (resp. \(\mathcal{P}_0(n,k)\)) that component of
\(\mathcal{F}(n,k)\) (resp. \(\mathcal{P}(n,k)\)) which contains the empty map. In the talk we will
present an essential complement to the above result. Namely, we will show
that the spaces \(\mathcal{F}_0(n,k)\) and \(\mathcal{P}_0(n,k)\) are not homotopy equivalent
for \(n>1\). Unfortunately, the problem in the case \(n=1\) remains unsolved.
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References-
P. Bartłomiejczyk, P. Nowak-Przygodzki,
The exponential law for partial, local and proper maps
and its application to otopy theory ,
to appear in Comm. Contemp. Math.
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P. Bartłomiejczyk, P. Nowak-Przygodzki,
On the homotopy equivalence
of the spaces of proper and local maps ,
Cent. Eur. J. Math. 12(9) 2014, 1330-1336.
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