Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod 2 cohomology (as a graded vector space, a ring and even an unstable algebra) but with all degrees divided by two, generalizing the classical examples of complex projective spaces under complex conjugation. Spaces which are constructed from unit balls in complex Euclidean spaces are called spherical and are very well understood. Our aim is twofold. We construct ‘exotic’ conjugation spaces and study the realization question: which spaces can be realized as real loci, i.e., fixed points of conjugation spaces. We identify obstructions and provide examples of spaces and manifolds which cannot be realized as such.

中文翻译：

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实现双打：共轭动物园

共轭空间是配备对合的拓扑空间，使得它们的不动点具有相同的 mod 2 上同调（作为分级向量空间、环甚至是不稳定代数），但所有度都除以 2，概括了复射影的经典例子复共轭下的空间。由复欧几里得空间中的单位球构成的空间称为球面，并且很好理解。我们的目标是双重的。我们构建“奇异”共轭空间并研究实现问题：哪些空间可以实现为真实轨迹，即共轭空间的不动点。我们识别障碍物并提供无法实现的空间和流形的示例。