Return to Question

Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(M)$ denotes the identity component of $\operatorname{Homeo}(M)$). More generally, we could let $M$ be compact with boundary and consider homeomorphisms fixing the boundary.

What known methods/invariants can be used to certify that a given element $\varphi\in\operatorname{MCG}(M)$ has infinite order?

Of course, $\varphi$ might have infinite order for homotopical reasons, but I am primarily interested in finer invariants. So, for the sake of this question, let's assume $\varphi$ is homotopic to the identity as maps $M\to M$. Of course, if $M$ has boundary, then this means homotopic through maps fixing the boundary.

For instance, the kernels in this answer are not detected by their action on homotopy, however unfortunately they are purely torsion (though I would still like to know how they are detected).

The focus of this question is on the high-dimensional case, though an answer in any dimension $\geq 4$ would be interesting.

Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(M)$ denotes the identity component of $\operatorname{Homeo}(M)$). More generally, we could let $M$ be compact with boundary and consider homeomorphisms fixing the boundary.

What known methods/invariants can be used to certify that a given element $\varphi\in\operatorname{MCG}(M)$ has infinite order?

Of course, $\varphi$ might have infinite order for homotopical reasons, but I am primarily interested in finer invariants. So, for the sake of this question, let's assume $\varphi$ is homotopic to the identity as maps $M\to M$. Of course, if $M$ has boundary, then this means homotopic through maps fixing the boundary.

For instance, the kernels in this answer are not detected by their action on homotopy, however unfortunately they are purely torsion (though I would still like to know how they are detected).

The focus of this question is on the high-dimensional case, though an answer in any dimension $\geq 4$ would be interesting.

Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(M)$ denotes the identity component of $\operatorname{Homeo}(M)$). More generally, we could let $M$ be compact with boundary and consider homeomorphisms fixing the boundary.

What known methods/invariants can be used to certify that a given element $\varphi\in\operatorname{MCG}(M)$ has infinite order?

Of course, $\varphi$ might have infinite order for homotopical reasons, but I am primarily interested in finer invariants. So, for the sake of this question, let's assume $\varphi$ is homotopic to the identity as maps $M\to M$ (necessarily through maps which are not homeomorphisms). For instance, the kernels in this answer are not detected by their homotopy classes, though unfortunately they appear to have no elements of infinite order (and anyway I don't know how they are detected).

The focus of this question is on the high-dimensional case, though an answer in any dimension $\geq 4$ would be interesting.

Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(M)$ denotes the identity component of $\operatorname{Homeo}(M)$). More generally, we could let $M$ be compact with boundary and consider homeomorphisms fixing the boundary.

What known methods/invariants can be used to certify that a given element $\varphi\in\operatorname{MCG}(M)$ has infinite order?

Of course, $\varphi$ might have infinite order for homotopical reasons, but I am primarily interested in finer invariants. So, for the sake of this question, let's assume $\varphi$ is homotopic to the identity as maps $M\to M$. Of course, if $M$ has boundary, then this means homotopic through maps fixing the boundary.

For instance, the kernels in this answer are not detected by their action on homotopy, however unfortunately they are purely torsion (though I would still like to know how they are detected).

The focus of this question is on the high-dimensional case, though an answer in any dimension $\geq 4$ would be interesting.