作为infinite-dimensional流形的自由拓扑逆半群
Free topological inverse semigroups as infinite-dimensional manifolds
摘要：
设$K$是拓扑逆Clifford半群的完备拟簇，包含所有拓扑半格.证明了$K$类中$X$的自由拓扑逆半群$F(X,K)$是$R^\infty$ -流形当且仅当$X$没有孤立点且$F(X,K)$是$R^\infty$ -流形的收缩元.我们由此得到：对于$R^\infty$ -流形的任何收缩$X$，自由拓扑逆半群$F(X,K)$是$R^\infty$ -流形当且仅当空间$X$没有孤立点.证明了对于无孤立点的$R^\infty$ -流形的任意同伦等价收缩$X,Y$，自由拓扑逆半群$F(X,K)$和$F(Y,K)$是同胚的.这允许我们构造非同胚空间，其自由拓扑逆半群是同胚的。
英文摘要：
Let $K$ be a complete quasivariety of topological inverse Clifford semigroups, containing all topological semilattices. It is shown that the free topological inverse semigroup $F(X,K)$ of $X$ in the class $K$ is an $R^\infty$-manifold if and only if $X$ has no isolated points and $F(X,K)$ is a retract of an $R^\infty$-manifold. We derive from this that for any retract $X$ of an $R^\infty$-manifold the free topological inverse semigroup $F(X,K)$ is an $R^\infty$-manifold if and only if the space $X$ has no isolated points. Also we show that for any homotopically equivalent retracts $X,Y$ of $R^\infty$-manifolds with no isolated points the free topological inverse semigroups $F(X,K)$ and $F(Y,K)$ are homeomorphic. This allows us to construct non-homeomorphic spaces whose free topological inverse semigroups are homeomorphic.
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