11 problems

Let $M$ be a simply connected compact oriented smooth manifold, let $LM=\operatorname{map}(S^1,M)$ be its free loop space, and write $\mathbb{H}_*(LM)=H_{*+d}(LM)$ for shifted free…

Bökstedt–Ottosen conjecture. The observed splitting is induced by an actual splitting of the suspension spectrum of $\Lambda M$ for these spaces.

Free-loop-space operations conjecture. These operations should extend to operations

Let $l\!k$ be a field and $X$ a simply connected finite CW complex. Write $LX=\operatorname{Map}(S^1,X)$ for the free loop space of $X$, and let $H^*(X;l\!k)$ denote its cohomology…

Let $X$ be a simply connected space and let $\mathcal{L}X=\operatorname{Map}(S^1,X)$ be its free loop space. Write $\Omega X$ for the based loop space, and call $X$ rationally hype…

Let $X$ be a finite type space of finite Lusternik–Schnirelmann category. It is rationally hyperbolic when the dimension of $\pi_\ast(X)\otimes\mathbb{Q}$ is infinite, and let…

Let $X$ be a closed manifold, and let $d4db(X)$ denote its free loop space. Its rational homology is $H_\ast(\mathcal{L}(X);\mathbb{Q})$. Gromov's conjecture. The rational homology…

Let $M$ be a simply-connected manifold, let $C^*(M)$ denote its differential graded algebra of singular cochains, and let $LM$ be its free loop space. Write $H^{>0}(LM)$ for the po…

Let $X$ be a simply-connected finite CW-complex, let ${\Bbbk}$ be a field, and let $i:\Omega X\rightarrow LX$ be the inclusion of based loops into free loops. Suppose that … is ont…

Let $X$ be a simply-connected finite CW-complex. The conjecture. Corollary 2 holds for any such $X$. This extends Corollary 2 from the setting established in the paper to all simpl…

Let $p$) be a prime. Write $L\mathbb{C}P^{p-1}$ for the free loop space of complex projective space and $\mathbb{H}_*(L\mathbb{C}P^{p-1};\mathbb{Z}_p)$ for its loop homology with c…