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

### Abstract

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.

### References

[1]

Noguchi H., “A generalization of absolute neighbourhood retracts,” Kodai Math. Sem. Rep., 5, No. 1, 20–22 (1953).

[2]

Wilansky A., Functional Analysis, Wiley, New York (1964).

[3]

Chernikov P. V., “Fixed points and ε-retracts [in Russian],” Mat. Zamet. YaGU, 17, No. 1, 146–148 (2010).

[4]

Chapman T., Lectures On Hilbert Cube Manifolds [in Russian], Mir, Moscow (1981).

[5]

Borsuk K., Theory of Retracts [in Russian], Mir, Moscow (1971).

[6]

Fedorchuk V. V. and Chigogidze A. Ch., Absolute Retracts and Infinite-Dimensional Manifolds [in Russian], Nauka, Moscow (1992).

[7]

Chernikov P. V., “Absolute σ-retracts and the Luzin theorem [in Russian],” Mat. Zamet. SVFU, 25, No. 2, 55–64 (2018).

[8]

Chernikov P. V., “Metric spaces and the continuation of mappings [in Russian],” Sib. Math. J., 27, No. 6, 210–215 (1986).

[9]

Chernikov P. V., “On the continuation of mappings [in Russian],” Sib. Math. J., 26, No. 2, 215–217 (1985).

[10]

Chernikov P. V., “On one characterization of absolute retracts [in Russian],” Sib. Math. J., 33, No. 2, 215–217 (1992).

[11]

Vinnik E. and Chernikov P. V., “On fixed points [in Russian],” Matematika v Shkole (to be published).

[12]

Edwards R., Functional Analysis [in Russian], Mir, Moscow (1969).

*Mathematical notes of NEFU*, 26(3), pp. 90-97. doi: https://doi.org/10.25587/SVFU.2019.89.45.007.

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