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Let $X$ be a cell complex obtained by attaching a 2-cell to a finite bouquet of circles (for example, a closed surface). In terms of the combinatorial type of the attaching map, the paper gives conditions for the existence of a fixed point free (topological) homeomorphism of the complex $X$. Also, quotients of finite group actions on such complexes are considered as well as a condition under which the induced actions on cohomology are trivial.
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