Mixing
Let $gammain S_{n+1}$ act naturally on $Delta^n$, then for some $sigma:Delta^nto X$,...
$begingroup$
Suppose we are given a topological space $X$, and a singular simplex $sigma:Delta^nto X$, where $Delta^n$ is the convex hull of $e_0,ldots,e_ninmathbb R^{n+1}$. For any permutation $gammain S_{n+1}$, we can define $gamma:Delta^ntoDelta^n$ by
$$
gammaleft(sum_{i=0}^n alpha_i e_iright)=sum_{i=0}^n alpha_i e_{gamma(i)}.
$$
Intuitively I would expect that we have $sigmacircgammasimoperatorname{sgn}(gamma)sigma$ in homology, i.e. $sigmacircgamma-operatorname{sgn}(gamma)sigma$ is a boundary. Clearly we can reduce to the case where $gamma$ is a swap, but I've struggled to find a clear proof of this statement even in this case, since I don't see such an obvious candidate for a $tauin C_{n+1}(X)$ such that $dtau=sigmacircgamma-operatorname{sgn}(gamma)sigma$.
algebraic-topology
asked Jan 21 at 10:42
Monstrous MoonshineMonstrous Moonshine
2,7511630
$endgroup$
add a comment |
$begingroup$
Suppose we are given a topological space $X$, and a singular simplex $sigma:Delta^nto X$, where $Delta^n$ is the convex hull of $e_0,ldots,e_ninmathbb R^{n+1}$. For any permutation $gammain S_{n+1}$, we can define $gamma:Delta^ntoDelta^n$ by
$$
gammaleft(sum_{i=0}^n alpha_i e_iright)=sum_{i=0}^n alpha_i e_{gamma(i)}.
$$
Intuitively I would expect that we have $sigmacircgammasimoperatorname{sgn}(gamma)sigma$ in homology, i.e. $sigmacircgamma-operatorname{sgn}(gamma)sigma$ is a boundary. Clearly we can reduce to the case where $gamma$ is a swap, but I've struggled to find a clear proof of this statement even in this case, since I don't see such an obvious candidate for a $tauin C_{n+1}(X)$ such that $dtau=sigmacircgamma-operatorname{sgn}(gamma)sigma$.
algebraic-topology
asked Jan 21 at 10:42
Monstrous MoonshineMonstrous Moonshine
2,7511630
$endgroup$
$begingroup$
Why do we even have an action on homology, i.e. why does the map on $C_n(X)$ induced by $gamma$ respect cycles and boundaries?
$endgroup$
– Lukas
Jan 23 at 17:08
add a comment |
$begingroup$
Suppose we are given a topological space $X$, and a singular simplex $sigma:Delta^nto X$, where $Delta^n$ is the convex hull of $e_0,ldots,e_ninmathbb R^{n+1}$. For any permutation $gammain S_{n+1}$, we can define $gamma:Delta^ntoDelta^n$ by
$$
gammaleft(sum_{i=0}^n alpha_i e_iright)=sum_{i=0}^n alpha_i e_{gamma(i)}.
$$
Intuitively I would expect that we have $sigmacircgammasimoperatorname{sgn}(gamma)sigma$ in homology, i.e. $sigmacircgamma-operatorname{sgn}(gamma)sigma$ is a boundary. Clearly we can reduce to the case where $gamma$ is a swap, but I've struggled to find a clear proof of this statement even in this case, since I don't see such an obvious candidate for a $tauin C_{n+1}(X)$ such that $dtau=sigmacircgamma-operatorname{sgn}(gamma)sigma$.
algebraic-topology
asked Jan 21 at 10:42
Monstrous MoonshineMonstrous Moonshine
2,7511630
$endgroup$
Suppose we are given a topological space $X$, and a singular simplex $sigma:Delta^nto X$, where $Delta^n$ is the convex hull of $e_0,ldots,e_ninmathbb R^{n+1}$. For any permutation $gammain S_{n+1}$, we can define $gamma:Delta^ntoDelta^n$ by
$$
gammaleft(sum_{i=0}^n alpha_i e_iright)=sum_{i=0}^n alpha_i e_{gamma(i)}.
$$
Intuitively I would expect that we have $sigmacircgammasimoperatorname{sgn}(gamma)sigma$ in homology, i.e. $sigmacircgamma-operatorname{sgn}(gamma)sigma$ is a boundary. Clearly we can reduce to the case where $gamma$ is a swap, but I've struggled to find a clear proof of this statement even in this case, since I don't see such an obvious candidate for a $tauin C_{n+1}(X)$ such that $dtau=sigmacircgamma-operatorname{sgn}(gamma)sigma$.
algebraic-topology
algebraic-topology
asked Jan 21 at 10:42
Monstrous MoonshineMonstrous Moonshine
2,7511630
asked Jan 21 at 10:42
Monstrous MoonshineMonstrous Moonshine
2,7511630
asked Jan 21 at 10:42
Monstrous MoonshineMonstrous Moonshine
2,7511630
asked Jan 21 at 10:42
Monstrous MoonshineMonstrous Moonshine
2,7511630
asked Jan 21 at 10:42
Monstrous MoonshineMonstrous Moonshine
2,7511630
2,7511630
$begingroup$
Why do we even have an action on homology, i.e. why does the map on $C_n(X)$ induced by $gamma$ respect cycles and boundaries?
$endgroup$
– Lukas
Jan 23 at 17:08
add a comment |
$begingroup$
Why do we even have an action on homology, i.e. why does the map on $C_n(X)$ induced by $gamma$ respect cycles and boundaries?
$endgroup$
– Lukas
Jan 23 at 17:08
$begingroup$
Why do we even have an action on homology, i.e. why does the map on $C_n(X)$ induced by $gamma$ respect cycles and boundaries?
$endgroup$
– Lukas
Jan 23 at 17:08
$begingroup$
Why do we even have an action on homology, i.e. why does the map on $C_n(X)$ induced by $gamma$ respect cycles and boundaries?
$endgroup$
– Lukas
Jan 23 at 17:08
add a comment |
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$begingroup$
Why do we even have an action on homology, i.e. why does the map on $C_n(X)$ induced by $gamma$ respect cycles and boundaries?
$endgroup$
– Lukas
Jan 23 at 17:08