Let $X$ be a nice space (manifold, CW-complex, what you prefer). I was wondering if there is a computable relation between the homology of $\Omega X$, the loop space of $X$, and the homology of $X$. I know that, almost by definition, the homotopy groups are the same (but shifted a dimension). Because the relation […]

Is it true that the loop space of a circle is contractible? Consider the long exact sequence in homotopy for the path fibration $\Omega S^1 \rightarrow \ast \rightarrow S^1$ shows all homotopy groups of the loop space to be zero, and then Whitehead’s theorem kicks in and tells us that $\Omega S^1$ is contractible. I […]

Every monoid $M$ is a category with one object $M$ and morphisms the elements of $M$. [Martin Brandenburg.] Every small category $C$ has a classifying space $BC$, defined as the geometric realization of the nerve. [Martin Brandenburg.] The classifying space $BM$ of a monoid $M$ is (by definition) the classifying space of the corresponding category.[Martin […]

I am not sure if there is an obvious answer to this, but this has been bothering me. Let $X$ be a topological space. When is the free loop space, $LX$, simply connected? Correct me if I’m wrong, but I believe that if $X$ is simply connected, then $LX$ is also simply connected since if […]

I am reading the wiki page on operad theory and I am trying to figure out how exactly those “Little something” operads work which are mentioned there. Specifically, I am having a hard time, despite the verbal statements on the page, grasping what is really going on with the n-discs operads (you have an illustration […]

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