Group cohomology topologically with simplicial sets
$begingroup$
I have a question about the usual formula for the differential in the usual projective resolution of $mathbb{Z}$ as a $G$-module for a finite group $G$
Recall that for a $G$-module $A$, $C^i(G,A)$ is defined as the abelian group of functions $G^ito A$ and the differential is $d(f)(g_0,...,g_i)=$ $$g_0cdot f(g_1,...,g_i) + displaystylesum_{j=1}^i(-1)^jf(g_0,...,g_{j-1}g_j, ..., g_i) + (-1)^{i+1}f(g_0,...,g_{i-1})$$
I was trying to find an interpretation for this formula, and for this I found out that there was a nice topological point of view on group cohomology. Pick a $K(G,1)$ space $X$, and let $p:tilde{X}to X$ be its universal covering space.
Then $pi_1(X)=G$ acts on $tilde{X}$ and thus on the singular complex $C_*(tilde{X})$, making $C_*(tilde{X})to mathbb{Z}$ a projective (actually free since the action on $C_*(tilde{X})$ comes from an action on the simplices and the action of $G$ on $tilde{X}$ is free) resolution of $mathbb{Z}$ as a trivial $G$-module ( the complex $C_*(tilde{X})to mathbb{Z}$ is exact because $tilde{X}$ is simply connected and has zero higher homotopy groups (because so does $X$), thus is contractible, because it's a CW-complex)
Now we also know that $|BG|$ is a $K(G,1)$ space, where $BG$ is a the nerve of $G$ (seen as a category), and we already know its universal cover, it's $|EG|$ (I don't know if it's standard notation, so let me make it clear) where $(EG)_n = G^{n+1}$ and $d_i(g_0,...,g_n) = (g_0,...,widehat{g_i},...,g_n)$ and $s_i(g_0,...,g_n) = (g_0,...,g_i,g_i,...,g_n)$; which gives another explicit cochain complex whose cohomology is $H^*(G,mathbb{Z})$. I had already seen this complex and now this gives me a topological interpretation for it.
But my trouble is with the first one I introduced. Indeed the formula for $d$ seems like a twisted version of $displaystylesum_{j=0}^i(-1)^j d_j$ where $d_j$ is the boundary of $BG$; twisted by the action of $G$ but only on the first summand. I'd like to understand this connection with BG (and not $|BG|$, that I understand, I think) more precisely than "it looks a bit similar":
How do we get this formula for $d$ from the nerve $BG$ ? What is the topological interpretation of this formula ?
algebraic-topology homology-cohomology group-cohomology simplicial-stuff
asked Dec 24 '18 at 12:12
MaxMax
13.3k11040
$endgroup$
|
show 1 more comment
$begingroup$
I have a question about the usual formula for the differential in the usual projective resolution of $mathbb{Z}$ as a $G$-module for a finite group $G$
Recall that for a $G$-module $A$, $C^i(G,A)$ is defined as the abelian group of functions $G^ito A$ and the differential is $d(f)(g_0,...,g_i)=$ $$g_0cdot f(g_1,...,g_i) + displaystylesum_{j=1}^i(-1)^jf(g_0,...,g_{j-1}g_j, ..., g_i) + (-1)^{i+1}f(g_0,...,g_{i-1})$$
I was trying to find an interpretation for this formula, and for this I found out that there was a nice topological point of view on group cohomology. Pick a $K(G,1)$ space $X$, and let $p:tilde{X}to X$ be its universal covering space.
Then $pi_1(X)=G$ acts on $tilde{X}$ and thus on the singular complex $C_*(tilde{X})$, making $C_*(tilde{X})to mathbb{Z}$ a projective (actually free since the action on $C_*(tilde{X})$ comes from an action on the simplices and the action of $G$ on $tilde{X}$ is free) resolution of $mathbb{Z}$ as a trivial $G$-module ( the complex $C_*(tilde{X})to mathbb{Z}$ is exact because $tilde{X}$ is simply connected and has zero higher homotopy groups (because so does $X$), thus is contractible, because it's a CW-complex)
Now we also know that $|BG|$ is a $K(G,1)$ space, where $BG$ is a the nerve of $G$ (seen as a category), and we already know its universal cover, it's $|EG|$ (I don't know if it's standard notation, so let me make it clear) where $(EG)_n = G^{n+1}$ and $d_i(g_0,...,g_n) = (g_0,...,widehat{g_i},...,g_n)$ and $s_i(g_0,...,g_n) = (g_0,...,g_i,g_i,...,g_n)$; which gives another explicit cochain complex whose cohomology is $H^*(G,mathbb{Z})$. I had already seen this complex and now this gives me a topological interpretation for it.
But my trouble is with the first one I introduced. Indeed the formula for $d$ seems like a twisted version of $displaystylesum_{j=0}^i(-1)^j d_j$ where $d_j$ is the boundary of $BG$; twisted by the action of $G$ but only on the first summand. I'd like to understand this connection with BG (and not $|BG|$, that I understand, I think) more precisely than "it looks a bit similar":
How do we get this formula for $d$ from the nerve $BG$ ? What is the topological interpretation of this formula ?
algebraic-topology homology-cohomology group-cohomology simplicial-stuff
asked Dec 24 '18 at 12:12
MaxMax
13.3k11040
$endgroup$
$begingroup$
Do you know about cohomology with local coefficients? (Or, better, sheaf cohomology?)
$endgroup$
– Moishe Cohen
Jan 6 at 17:03
$begingroup$
@MoisheCohen I know the definition of sheaf cohomology and some of its applications but that's about it; and the same for cohomology with local coefficients. But since I'm only looking for some interpretation of these formulas, some motivation, maybe it's enough ?
$endgroup$
– Max
Jan 6 at 17:07
$begingroup$
The answer to your question lies in the cohomology with local coefficients $H^*(BG, V)$, where $V$ is the flat bundle (local system) associated with the $G$-module $A$. You have to think about simplicial cohomology of this local system. When you write down the cochain complex for this, you will discover the coboundary formula in the group cohomology. The "twisting" comes from the fact that the local system is (in general) nontrivial.
$endgroup$
– Moishe Cohen
Jan 6 at 17:11
$begingroup$
@MoisheCohen : when you write $BG$ you mean what I called $|BG|$ or is there some notion of cohomology with local coefficients for simplicial sets (that I don't know about ) ? And if yes, what's the flat bundle associated with $A$ ? Is it just "$A$ on objects and identifying (in a fixed way) $hom(x,y)$ with $G$ for all $x,y$, we get the action on maps" ? *
$endgroup$
– Max
Jan 6 at 17:34
$begingroup$
@Max Since $A$ is a $G$-module, there is a natural map $G to text{End}(A)$. As $G = pi_1 BG$, this gives a representation $pi_1 BG to text{End}(A)$. Consider the locally constant $A$-valued sheaf on $BG$ corresponding to this representation - that's the associated flat bundle. You can alternatively describe this as $(EG times A)/G to BG$ where $G$ acts diagonally on $EG times A$.
$endgroup$
– Balarka Sen
Jan 6 at 20:21
|
show 1 more comment
$begingroup$
I have a question about the usual formula for the differential in the usual projective resolution of $mathbb{Z}$ as a $G$-module for a finite group $G$
Recall that for a $G$-module $A$, $C^i(G,A)$ is defined as the abelian group of functions $G^ito A$ and the differential is $d(f)(g_0,...,g_i)=$ $$g_0cdot f(g_1,...,g_i) + displaystylesum_{j=1}^i(-1)^jf(g_0,...,g_{j-1}g_j, ..., g_i) + (-1)^{i+1}f(g_0,...,g_{i-1})$$
I was trying to find an interpretation for this formula, and for this I found out that there was a nice topological point of view on group cohomology. Pick a $K(G,1)$ space $X$, and let $p:tilde{X}to X$ be its universal covering space.
Then $pi_1(X)=G$ acts on $tilde{X}$ and thus on the singular complex $C_*(tilde{X})$, making $C_*(tilde{X})to mathbb{Z}$ a projective (actually free since the action on $C_*(tilde{X})$ comes from an action on the simplices and the action of $G$ on $tilde{X}$ is free) resolution of $mathbb{Z}$ as a trivial $G$-module ( the complex $C_*(tilde{X})to mathbb{Z}$ is exact because $tilde{X}$ is simply connected and has zero higher homotopy groups (because so does $X$), thus is contractible, because it's a CW-complex)
Now we also know that $|BG|$ is a $K(G,1)$ space, where $BG$ is a the nerve of $G$ (seen as a category), and we already know its universal cover, it's $|EG|$ (I don't know if it's standard notation, so let me make it clear) where $(EG)_n = G^{n+1}$ and $d_i(g_0,...,g_n) = (g_0,...,widehat{g_i},...,g_n)$ and $s_i(g_0,...,g_n) = (g_0,...,g_i,g_i,...,g_n)$; which gives another explicit cochain complex whose cohomology is $H^*(G,mathbb{Z})$. I had already seen this complex and now this gives me a topological interpretation for it.
But my trouble is with the first one I introduced. Indeed the formula for $d$ seems like a twisted version of $displaystylesum_{j=0}^i(-1)^j d_j$ where $d_j$ is the boundary of $BG$; twisted by the action of $G$ but only on the first summand. I'd like to understand this connection with BG (and not $|BG|$, that I understand, I think) more precisely than "it looks a bit similar":
How do we get this formula for $d$ from the nerve $BG$ ? What is the topological interpretation of this formula ?
algebraic-topology homology-cohomology group-cohomology simplicial-stuff
asked Dec 24 '18 at 12:12
MaxMax
13.3k11040
$endgroup$
I have a question about the usual formula for the differential in the usual projective resolution of $mathbb{Z}$ as a $G$-module for a finite group $G$
Recall that for a $G$-module $A$, $C^i(G,A)$ is defined as the abelian group of functions $G^ito A$ and the differential is $d(f)(g_0,...,g_i)=$ $$g_0cdot f(g_1,...,g_i) + displaystylesum_{j=1}^i(-1)^jf(g_0,...,g_{j-1}g_j, ..., g_i) + (-1)^{i+1}f(g_0,...,g_{i-1})$$
I was trying to find an interpretation for this formula, and for this I found out that there was a nice topological point of view on group cohomology. Pick a $K(G,1)$ space $X$, and let $p:tilde{X}to X$ be its universal covering space.
Then $pi_1(X)=G$ acts on $tilde{X}$ and thus on the singular complex $C_*(tilde{X})$, making $C_*(tilde{X})to mathbb{Z}$ a projective (actually free since the action on $C_*(tilde{X})$ comes from an action on the simplices and the action of $G$ on $tilde{X}$ is free) resolution of $mathbb{Z}$ as a trivial $G$-module ( the complex $C_*(tilde{X})to mathbb{Z}$ is exact because $tilde{X}$ is simply connected and has zero higher homotopy groups (because so does $X$), thus is contractible, because it's a CW-complex)
Now we also know that $|BG|$ is a $K(G,1)$ space, where $BG$ is a the nerve of $G$ (seen as a category), and we already know its universal cover, it's $|EG|$ (I don't know if it's standard notation, so let me make it clear) where $(EG)_n = G^{n+1}$ and $d_i(g_0,...,g_n) = (g_0,...,widehat{g_i},...,g_n)$ and $s_i(g_0,...,g_n) = (g_0,...,g_i,g_i,...,g_n)$; which gives another explicit cochain complex whose cohomology is $H^*(G,mathbb{Z})$. I had already seen this complex and now this gives me a topological interpretation for it.
But my trouble is with the first one I introduced. Indeed the formula for $d$ seems like a twisted version of $displaystylesum_{j=0}^i(-1)^j d_j$ where $d_j$ is the boundary of $BG$; twisted by the action of $G$ but only on the first summand. I'd like to understand this connection with BG (and not $|BG|$, that I understand, I think) more precisely than "it looks a bit similar":
How do we get this formula for $d$ from the nerve $BG$ ? What is the topological interpretation of this formula ?
algebraic-topology homology-cohomology group-cohomology simplicial-stuff
algebraic-topology homology-cohomology group-cohomology simplicial-stuff
asked Dec 24 '18 at 12:12
MaxMax
13.3k11040
asked Dec 24 '18 at 12:12
MaxMax
13.3k11040
asked Dec 24 '18 at 12:12
MaxMax
13.3k11040
asked Dec 24 '18 at 12:12
MaxMax
13.3k11040
asked Dec 24 '18 at 12:12
MaxMax
13.3k11040
13.3k11040
$begingroup$
Do you know about cohomology with local coefficients? (Or, better, sheaf cohomology?)
$endgroup$
– Moishe Cohen
Jan 6 at 17:03
$begingroup$
@MoisheCohen I know the definition of sheaf cohomology and some of its applications but that's about it; and the same for cohomology with local coefficients. But since I'm only looking for some interpretation of these formulas, some motivation, maybe it's enough ?
$endgroup$
– Max
Jan 6 at 17:07
$begingroup$
The answer to your question lies in the cohomology with local coefficients $H^*(BG, V)$, where $V$ is the flat bundle (local system) associated with the $G$-module $A$. You have to think about simplicial cohomology of this local system. When you write down the cochain complex for this, you will discover the coboundary formula in the group cohomology. The "twisting" comes from the fact that the local system is (in general) nontrivial.
$endgroup$
– Moishe Cohen
Jan 6 at 17:11
$begingroup$
@MoisheCohen : when you write $BG$ you mean what I called $|BG|$ or is there some notion of cohomology with local coefficients for simplicial sets (that I don't know about ) ? And if yes, what's the flat bundle associated with $A$ ? Is it just "$A$ on objects and identifying (in a fixed way) $hom(x,y)$ with $G$ for all $x,y$, we get the action on maps" ? *
$endgroup$
– Max
Jan 6 at 17:34
$begingroup$
@Max Since $A$ is a $G$-module, there is a natural map $G to text{End}(A)$. As $G = pi_1 BG$, this gives a representation $pi_1 BG to text{End}(A)$. Consider the locally constant $A$-valued sheaf on $BG$ corresponding to this representation - that's the associated flat bundle. You can alternatively describe this as $(EG times A)/G to BG$ where $G$ acts diagonally on $EG times A$.
$endgroup$
– Balarka Sen
Jan 6 at 20:21
|
show 1 more comment
$begingroup$
Do you know about cohomology with local coefficients? (Or, better, sheaf cohomology?)
$endgroup$
– Moishe Cohen
Jan 6 at 17:03
$begingroup$
@MoisheCohen I know the definition of sheaf cohomology and some of its applications but that's about it; and the same for cohomology with local coefficients. But since I'm only looking for some interpretation of these formulas, some motivation, maybe it's enough ?
$endgroup$
– Max
Jan 6 at 17:07
$begingroup$
The answer to your question lies in the cohomology with local coefficients $H^*(BG, V)$, where $V$ is the flat bundle (local system) associated with the $G$-module $A$. You have to think about simplicial cohomology of this local system. When you write down the cochain complex for this, you will discover the coboundary formula in the group cohomology. The "twisting" comes from the fact that the local system is (in general) nontrivial.
$endgroup$
– Moishe Cohen
Jan 6 at 17:11
$begingroup$
@MoisheCohen : when you write $BG$ you mean what I called $|BG|$ or is there some notion of cohomology with local coefficients for simplicial sets (that I don't know about ) ? And if yes, what's the flat bundle associated with $A$ ? Is it just "$A$ on objects and identifying (in a fixed way) $hom(x,y)$ with $G$ for all $x,y$, we get the action on maps" ? *
$endgroup$
– Max
Jan 6 at 17:34
$begingroup$
@Max Since $A$ is a $G$-module, there is a natural map $G to text{End}(A)$. As $G = pi_1 BG$, this gives a representation $pi_1 BG to text{End}(A)$. Consider the locally constant $A$-valued sheaf on $BG$ corresponding to this representation - that's the associated flat bundle. You can alternatively describe this as $(EG times A)/G to BG$ where $G$ acts diagonally on $EG times A$.
$endgroup$
– Balarka Sen
Jan 6 at 20:21
$begingroup$
Do you know about cohomology with local coefficients? (Or, better, sheaf cohomology?)
$endgroup$
– Moishe Cohen
Jan 6 at 17:03
$begingroup$
Do you know about cohomology with local coefficients? (Or, better, sheaf cohomology?)
$endgroup$
– Moishe Cohen
Jan 6 at 17:03
$begingroup$
@MoisheCohen I know the definition of sheaf cohomology and some of its applications but that's about it; and the same for cohomology with local coefficients. But since I'm only looking for some interpretation of these formulas, some motivation, maybe it's enough ?
$endgroup$
– Max
Jan 6 at 17:07
$begingroup$
@MoisheCohen I know the definition of sheaf cohomology and some of its applications but that's about it; and the same for cohomology with local coefficients. But since I'm only looking for some interpretation of these formulas, some motivation, maybe it's enough ?
$endgroup$
– Max
Jan 6 at 17:07
$begingroup$
The answer to your question lies in the cohomology with local coefficients $H^*(BG, V)$, where $V$ is the flat bundle (local system) associated with the $G$-module $A$. You have to think about simplicial cohomology of this local system. When you write down the cochain complex for this, you will discover the coboundary formula in the group cohomology. The "twisting" comes from the fact that the local system is (in general) nontrivial.
$endgroup$
– Moishe Cohen
Jan 6 at 17:11
$begingroup$
The answer to your question lies in the cohomology with local coefficients $H^*(BG, V)$, where $V$ is the flat bundle (local system) associated with the $G$-module $A$. You have to think about simplicial cohomology of this local system. When you write down the cochain complex for this, you will discover the coboundary formula in the group cohomology. The "twisting" comes from the fact that the local system is (in general) nontrivial.
$endgroup$
– Moishe Cohen
Jan 6 at 17:11
$begingroup$
@MoisheCohen : when you write $BG$ you mean what I called $|BG|$ or is there some notion of cohomology with local coefficients for simplicial sets (that I don't know about ) ? And if yes, what's the flat bundle associated with $A$ ? Is it just "$A$ on objects and identifying (in a fixed way) $hom(x,y)$ with $G$ for all $x,y$, we get the action on maps" ? *
$endgroup$
– Max
Jan 6 at 17:34
$begingroup$
@MoisheCohen : when you write $BG$ you mean what I called $|BG|$ or is there some notion of cohomology with local coefficients for simplicial sets (that I don't know about ) ? And if yes, what's the flat bundle associated with $A$ ? Is it just "$A$ on objects and identifying (in a fixed way) $hom(x,y)$ with $G$ for all $x,y$, we get the action on maps" ? *
$endgroup$
– Max
Jan 6 at 17:34
$begingroup$
@Max Since $A$ is a $G$-module, there is a natural map $G to text{End}(A)$. As $G = pi_1 BG$, this gives a representation $pi_1 BG to text{End}(A)$. Consider the locally constant $A$-valued sheaf on $BG$ corresponding to this representation - that's the associated flat bundle. You can alternatively describe this as $(EG times A)/G to BG$ where $G$ acts diagonally on $EG times A$.
$endgroup$
– Balarka Sen
Jan 6 at 20:21
$begingroup$
@Max Since $A$ is a $G$-module, there is a natural map $G to text{End}(A)$. As $G = pi_1 BG$, this gives a representation $pi_1 BG to text{End}(A)$. Consider the locally constant $A$-valued sheaf on $BG$ corresponding to this representation - that's the associated flat bundle. You can alternatively describe this as $(EG times A)/G to BG$ where $G$ acts diagonally on $EG times A$.
$endgroup$
– Balarka Sen
Jan 6 at 20:21
|
show 1 more comment
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$begingroup$
Do you know about cohomology with local coefficients? (Or, better, sheaf cohomology?)
$endgroup$
– Moishe Cohen
Jan 6 at 17:03
$begingroup$
@MoisheCohen I know the definition of sheaf cohomology and some of its applications but that's about it; and the same for cohomology with local coefficients. But since I'm only looking for some interpretation of these formulas, some motivation, maybe it's enough ?
$endgroup$
– Max
Jan 6 at 17:07
$begingroup$
The answer to your question lies in the cohomology with local coefficients $H^*(BG, V)$, where $V$ is the flat bundle (local system) associated with the $G$-module $A$. You have to think about simplicial cohomology of this local system. When you write down the cochain complex for this, you will discover the coboundary formula in the group cohomology. The "twisting" comes from the fact that the local system is (in general) nontrivial.
$endgroup$
– Moishe Cohen
Jan 6 at 17:11
$begingroup$
@MoisheCohen : when you write $BG$ you mean what I called $|BG|$ or is there some notion of cohomology with local coefficients for simplicial sets (that I don't know about ) ? And if yes, what's the flat bundle associated with $A$ ? Is it just "$A$ on objects and identifying (in a fixed way) $hom(x,y)$ with $G$ for all $x,y$, we get the action on maps" ? *
$endgroup$
– Max
Jan 6 at 17:34
$begingroup$
@Max Since $A$ is a $G$-module, there is a natural map $G to text{End}(A)$. As $G = pi_1 BG$, this gives a representation $pi_1 BG to text{End}(A)$. Consider the locally constant $A$-valued sheaf on $BG$ corresponding to this representation - that's the associated flat bundle. You can alternatively describe this as $(EG times A)/G to BG$ where $G$ acts diagonally on $EG times A$.
$endgroup$
– Balarka Sen
Jan 6 at 20:21