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Let $LX$ denote the free loop space on $X$. We have an evaluation map $ev\colon LX\to X$ and we have an inclusion $X\hookrightarrow LX$ (where $x\in X$ is mapped to the constant loop at $x$). Suppose …

Let $M$ be a connected compact topological $n$-dimensional manifold without a boundary and with a CW-structure $M= \bigcup M^i$. We have that $$ (\#^g M)\smallsetminus D^n \simeq \bigvee_{i=1}^gM^{n- …

Let $H$ be a topological group which is a subgroup of two other topological groups $G$ and $G'$. It follows (from Rmk 8.9 in May - Classifying spaces and fibrations (MSN, free)) that there exist weak …

$CW_0$-complexes are analogs of $CW$-complexes, in which the "building blocks" are the rational disks $D^{n+1}_0$ whose boundaries are given by $\partial D^{n+1}_0= S^n_0$, where $S^n_0$ is a rational …

Let $A_m =(A,0,m_2,m_3,\dots)$ and $A_n=(A,0,n_2,n_3,\dots)$ be two $A_\infty$-structures on a vector space $A$. Assume that i) $A_m$ and $A_n$ are isomorphic, and ii) $A_m$ and $A_n$ have the same …

It is not completely true that the set of gauge equivalence classes of Maurer-Cartan elements of a dg Lie algebra is invariant under quasi-isomorphisms. Here is an example (which I learned from Profes …