$\varepsilon$-retracts, Q-manifolds, and fixed points

  • Chernikov Pavel V., smz@math.nsc.ru Novosibirsk State University, 1 Pirogov Street, Novosibirsk 630090
Keywords: $\varepsilon$-retract, Q-manifold, fixed point


In the class of compact metric spaces, we define the absolute neighborhood $\delta$-retract ($ANR_\varepsilon$-space), a generalization of the corresponding notion from [1]. One of the Noguchi fixed point theorems is generalized with the use of $ANR_\varepsilon$-space, from which the classic Brouwer fixed-point theorem follows. It is proved that there exists a compact noncollapsible acyclic $Q$-manifold with the fixed point property, while it was previously known that a compact non-acyclic $Q$-manifold existed. We prove that if the Cartesian product of a sequence of non-singleton compact metric sets belongs to $ANR$,
then it is a compact $Q$-manifold. A topological space with the fixed point $\sigma$-property is introduced and studied. In particular, we prove that an $ANR$-compact $X$ has the fixed point $\sigma$-property if and only if $X$ is connected. An example of a noncompact set in $R^2$ with the fixed point property is given.


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How to Cite
Chernikov, P. (2019) “$\varepsilon$-retracts, Q-manifolds, and fixed points”, Mathematical notes of NEFU, 26(3), pp. 90-97. doi: https://doi.org/10.25587/SVFU.2019.89.45.007.