Background
Data Presentation
Moduli spaces
This page contains data from the paper Configuration spaces on a wedge of spheres and higher Hochschild homology, by Nir Gadish and Louis Hainaut. Please check out our paper for more details.
Let \(X = \bigvee_{i=1}^g S^1\) be a wedge sum of \(g\) circles. Our work found universal \(\mathfrak{S}_n\times \mathfrak{S}_m\)-representations \(\Phi^p[n,m]\), such that the associated graded of the cohomology with compact support of configuration spaces on \(X\) decomposes as \(\operatorname{gr}^{\ast} H_c^i(F(X, n)) \cong \bigoplus_{p=0}^i{\Phi^p[n, i-p]\otimes_{\mathfrak{S}_{i-p}}\tilde{H}^{\ast}(X)^{\otimes i-p}}.\) This decomposition is actually valid for \(X\) any finite wedge of spheres of any dimensions. When \(X\) is a wedge of circles, the cohomology group \(H_c^i(F(X, n))\) is trivial unless \(i=n\) or \(i = n-1\); these cases will be called here respectively the codimension 0 and the codimension 1 case. The coefficients \(\Phi^p[n,m]\) decompose as a direct sum of \(\Phi^p[\lambda, \mu]\otimes (\chi_{\lambda}\boxtimes\chi_{\mu})\) over all partitions \(\lambda\vdash n\) and \(\mu\vdash m\) and these pieces are what is computed here.
The isotypical components of the cohomology groups \(H_c^n(F(X,n))\) and \(H_c^{n-1}(F(X,n))\) correspond respectively the the bead representations \(U_{\lambda}^{I}\) and \(U_{\lambda}^{II}\) defined by Turchin-Willwacher. Also, by work of Powell-Vespa, the codimension 0 case \(\bigoplus_{p=0}^n{\Phi^p[\lambda, n-p]\otimes_{\mathfrak{S}_{n-p}}(\mathbb{Q}^g)^{\otimes n-p}}\) gives the injective envelope \(\omega\beta_d S_{\lambda^{\dagger}}\) in the category of outer functors of the Schur functor \(S_{\lambda^{\dagger}}\) associated with the transposed partition \(\lambda^{\dagger}\) of \(\lambda\).
Our results also provide information about the compactly supported cohomology groups of \(\mathcal{M}_{2,n}\) in weight 0. These cohomology groups are nonzero only in degrees \(n+3, n+2\) (called respectively the codimension 0 and codimension 1 case). We precomputed the result up to \(n=17\) and provide ways of visualizing the result in the last part of this page.
In the section Data Presentation we present our results about the compactly supported cohomology of configuration spaces on a wedge of circles. If you are only interested in the cohomology of \(\mathcal{M}_{2,n}\) you should go directly to Moduli spaces. Keep in mind that all our results presented here are exact up to \(n=10\), and only provide a lower bound beyond.
The cell below contains the functions necessary to compute the cohomology, as well as some functions to help the presentation of the results. (These cells are linked, so please always start with evaluating the first one.)
Available functions are
Cohomology_ConfSpace(n, dim=1)
This function produces the decomposition of the cohomology of configuration spaces. It should always be called first when running the cell, but if you run the cell multiple times without changing the value for n nor for dim you can comment the line after the first time (by placing a # in front of the line) to avoid unnecessary calculations.
As the value of n increases, the computations take longer to finish. For n=10 it takes approximatively 30 seconds; it is not advised to try values n>13.
n, integer, denotes the number of points in the configuration space.
dim, integer, takes the value 1 by default. You can give it the value 2 (or any other even value) if you want to look at even (equi-dimensional) spheres. The default value dim=1 gives access to the coefficients \(\Phi^p[\lambda,\mu^*]\) (with the partition \(\mu\) conjugated). In order to obtain \(\Phi^p[\lambda,\mu]\) it is necessary to give the value dim=2 instead (or remember to conjugate the partition).
Focus_Cohomology(Hom, partition, Focus = "Sym", codim = 1, Filtered = False)
This function gives the isotypical component of the cohomology of configuration spaces corresponding to the given partition, representing either a \(S_n\)-representation or a \(GL\) representation.
Hom, symmetric function in two variables, encoding the simultaneous action of \(S_n\) and of \(GL\). Use the value computed at the beginning of the cell.
partition, integer partition. The sum of its parts must be equal to \(n\) if the variable Focus assumes the value "Sym", if Focus assumes the value "GL" this sum must be at most \(n-\)codim.
Focus, string, can only take one of the two values "Sym" or "GL". In the first case it outputs the \(GL\)-multiplicity of the \(S_n\)-representation associated to the given partition. In the second case it outputs the \(S_n\) multiplicity of the Schur functor associated with the given partition.
codim, integer, can only take the value 0 or 1. In the first case it outputs the cohomology \(H_c^n(F(X, n))\). In the second case it outputs the cohomology \(H_c^{n-1}(F(X,n))\).
Filtered, boolean, only matters when the variable Focus assumes the value "Sym". If set to False the function outputs a symmetric function. If set to True the function outputs a list of symmetric functions corresponding to each piece of the polynomial filtration.
Forget_Equivariance(Hom, Forget = "Sym", genus = 0, codim = 1, Filtered = False)
This function gives the cohomology of configuration spaces decomposed according to only one of the two actions of \(S_n\) and \(GL\).
Hom, symmetric function in two variables, encoding the simultaneous action of \(S_n\) and of \(GL\). Use the value computed at the beginning of the cell.
Forget, string, can only take one of the two values "Sym" or "GL". In the first case it replaces all \(S_n\) multiplicities by their dimension, in the second case it replaces all the \(GL\) multiplicities by their dimension.
genus, integer. Only necessary when Forget assumes the value "GL". Indicates the number of spheres of the space \(X\).
codim, integer, can only take the value 0 or 1. In the first case it outputs the cohomology \(H_c^n(F(X, n))\). In the second case it outputs the cohomology \(H_c^{n-1}(F(X,n))\).
Filtered, boolean. If set to False the function outputs a symmetric function. If set to True the function outputs a list of symmetric functions corresponding to each piece of the polynomial filtration.
Compute_Trace(Hom, evalues, partition, codim = 1, Filtered = False)
This function computes the trace under the \(GL\)-action of a diagonal matrix on our cohomology groups.
Hom, symmetric function in two variables, encoding the simultaneous action of \(S_n\) and of \(GL\). Use the value computed at the beginning of the cell.
evalues, list. Eigenvalues of the diagonal matrix to be evaluated.
partition, integer partition; the sum of its parts must be equal to \(n\). Outputs the trace of the corresponding \(S_n\) partition.
codim, integer, can only take the value 0 or 1. In the first case it outputs the cohomology \(H_c^n(F(X, n))\). In the second case it outputs the cohomology \(H_c^{n-1}(F(X,n))\).
Filtered, boolean. If set to False the function outputs a symmetric function. If set to True the function outputs a list of symmetric functions corresponding to each piece of the polynomial filtration.
The cell below shows our computations for the cohomology of \(\mathcal{M}_{2,n}\) in weight 0. You will need to evaluate the cell above the first time in order to have the functions defined, otherwise this cell can work from the ones in the previous section. It provides the following function:
Cohomology_M2n(points, Hom = None, codim = 1)
This function gives the cohomology of the moduli space \(\mathcal{M}_{2,n}\). The result is exact up to 10 points. For values between 11 and 17 it is only a lower bound.
points, integer, number of marked points.
Hom, Cohomology of configurations on a wedge of spheres. If set to None the function utilizes precomputed value. If you want to give a value you should compute it as in the previous section.
codim, integer, can only take the value 0 or 1. In the first case it outputs the cohomology \(H_c^{n+3}(\mathcal{M}_{2,n})\). In the second case it outputs the cohomology \(H_c^{n+2}(\mathcal{M}_{2,n})\).
You may also use the following function:
dimension(SymFunc, n)
This function gives the dimension of the degree \(n\) piece of a symmetric function.
SymFunc, symmetrif function, only the degree n part of this symmetric function matters.
n, integer. Degree in which we want to compute the dimension.