Return to Question

Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$. Let $$ {\cal D} \subset U(n) $$ denote the subspace of those matrices having a non-trivial $(+1)$-eigenspace.

Background: It is known that $\cal D$ has vanishing homology in dimension $n^2$. It is also not difficult to show that $H_{n^2-1}({\cal D}) \cong \Bbb Z$. Furthermore, it is known that ${\cal D}$ has the structure of a finite CW complex of dimension $n^2-1$.

For $g\in {\cal D}$, define the local homology by $$ H_{n^2-1}({\cal D}\, |\, g;\Bbb Q) := H_{n^2-1}({\cal D},{\cal D} \setminus g;\Bbb Q)\, . $$

Question: For arbitrary $g\in {\cal D}$ is the rank of this group known?

If yes, can anyone provide a reference?

Remark: Based on a direct computation when $n=2$, it seems reasonable to conjecture that the rank of $H_{n^2-1}({\cal D}\, |\, g;\Bbb Q)$ is equal to the dimension of the $(+1)$-eigenspace of $g$ (i.e., the multiplicity of the eigenvalue +1).

Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$). Let $$ {\cal D} \subset U(n) $$ denote the subspace of those matrices having a non-trivial $(+1)$-eigenspace.

Background: It is known that $\cal D$ has vanishing homology in dimension $n^2$. It is also not difficult to show that $H_{n^2-1}({\cal D}) \cong \Bbb Z$. Furthermore, it is known that ${\cal D}$ has the structure of a finite CW complex of dimension $n^2-1$.

For $g\in {\cal D}$, define the local homology by $$ H_{n^2-1}({\cal D}\, |\, g;\Bbb Q) := H_{n^2-1}({\cal D},{\cal D} \setminus g;\Bbb Q)\, . $$

Question: For arbitrary $g\in {\cal D}$ is the rank of this group known?

If yes, can anyone provide a reference?

Remark: Based on a direct computation when $n=2$, it seems reasonable to conjecture that the rank of $H_{n^2-1}({\cal D}\, |\, g;\Bbb Q)$ is equal to the dimension of the $(+1)$-eigenspace of $g$ (i.e., the multiplicity of the eigenvalue +1).

Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$. Let $$ {\cal D} \subset U(n) $$ denote the subspace of those matrices $A$ such that such that $A$ has a non-trivial $(+1)$-eigenspace. Clearly, $\cal D$ has vanishing homology in dimension $n^2$. It is not difficult to show that $H_{n^2-1}({\cal D}) \cong \Bbb Z$. (In fact, it is known that ${\cal D}$ has the structure of a finite CW complex of dimension $n^2-1$.)

For $g\in {\cal D}$, define the local homology by $$ H_{n^2-1}({\cal D}\, |\, g;\Bbb Q) := H_{n^2-1}({\cal D},{\cal D} \setminus g;\Bbb Q)\, . $$

Question: Is the rank of this group known?

If yes, can anyone provide a reference?

Remark: Based on a direct computation when $n=2$, it seems reasonable to conjecture that the rank of $H_{n^2-1}({\cal D}\, |\, g;\Bbb Q)$ is equal to the dimension of the $(+1)$-eigenspace of $g$ (i.e., the multiplicity of the eigenvalue +1).

Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$. Let $$ {\cal D} \subset U(n) $$ denote the subspace of those matrices having a non-trivial $(+1)$-eigenspace.

Background: It is known that $\cal D$ has vanishing homology in dimension $n^2$. It is also not difficult to show that $H_{n^2-1}({\cal D}) \cong \Bbb Z$. Furthermore, it is known that ${\cal D}$ has the structure of a finite CW complex of dimension $n^2-1$.

For $g\in {\cal D}$, define the local homology by $$ H_{n^2-1}({\cal D}\, |\, g;\Bbb Q) := H_{n^2-1}({\cal D},{\cal D} \setminus g;\Bbb Q)\, . $$

Question: For arbitrary $g\in {\cal D}$ is the rank of this group known?

If yes, can anyone provide a reference?

Remark: Based on a direct computation when $n=2$, it seems reasonable to conjecture that the rank of $H_{n^2-1}({\cal D}\, |\, g;\Bbb Q)$ is equal to the dimension of the $(+1)$-eigenspace of $g$ (i.e., the multiplicity of the eigenvalue +1).

Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$). Let $$ {\cal D} \subset U(n) $$ denote the subspace of those matrices having a non-trivial $(+1)$-eigenspace. It is not hard to show that $\cal D$ has vanishing homology in dimension $n^2$.

  It is also not difficult to show that $H_{n^2-1}({\cal D}) \cong \Bbb Z$. (In fact, it is known that ${\cal D}$ has the structure of a finite CW complex of dimension $n^2-1$.)

For $g\in {\cal D}$, define the local homology by $$ H_{n^2-1}({\cal D}\, |\, g;\Bbb Q) := H_{n^2-1}({\cal D},{\cal D} \setminus g;\Bbb Q)\, . $$

Question: Is the rank of this group known?

If yes, can anyone provide a reference?

Remark: Based on a direct computation when $n=2$, it seems reasonable to conjecture that the rank of $H_{n^2-1}({\cal D}\, |\, g;\Bbb Q)$ is equal to the dimension of the $(+1)$-eigenspace of $g$ (i.e., the multiplicity of the eigenvalue +1.

Let $U(n)$ denote the unitary group (this is a manifold of dimension $n^2$. Let $$ {\cal D} \subset U(n) $$ denote the subspace of those matrices $A$ such that such that $A$ has a non-trivial $(+1)$-eigenspace. Clearly, $\cal D$ has vanishing homology in dimension $n^2$. It is not difficult to show that $H_{n^2-1}({\cal D}) \cong \Bbb Z$. (In fact, it is known that ${\cal D}$ has the structure of a finite CW complex of dimension $n^2-1$.)

For $g\in {\cal D}$, define the local homology by $$ H_{n^2-1}({\cal D}\, |\, g;\Bbb Q) := H_{n^2-1}({\cal D},{\cal D} \setminus g;\Bbb Q)\, . $$

Question: Is the rank of this group known?

If yes, can anyone provide a reference?

Remark: Based on a direct computation when $n=2$, it seems reasonable to conjecture that the rank of $H_{n^2-1}({\cal D}\, |\, g;\Bbb Q)$ is equal to the dimension of the $(+1)$-eigenspace of $g$ (i.e., the multiplicity of the eigenvalue +1).