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May 11, 2017 at 23:00 vote accept John Klein
S May 11, 2017 at 22:06 history bounty ended John Klein
S May 11, 2017 at 22:06 history notice removed John Klein
May 7, 2017 at 1:42 vote accept John Klein
May 11, 2017 at 22:06
May 6, 2017 at 8:23 comment added Mark Grant @JohnKlein: I don't know yet. Two unitary matrices are unitarily similar iff they have the same eigenvalues, so that map is actually the orbit map of the conjugation action, and $\mathcal{D}$ is the preimage of the inclusion $SP^{n-1}(S^1)\subset SP^n(S^1)$ obtained by adding a 1. Surely its not a bundle though. Greg's answer seems to be more a propos.
May 5, 2017 at 21:59 answer added Gregory Arone timeline score: 11
May 5, 2017 at 16:05 comment added John Klein @MarkGrant I am aware of that map. It can be used to equip $U(n)$ with the structure of a Whitney stratified space (I learned this from David Ayala). But how can you use the map to compute the local homology of $\cal D$?
May 5, 2017 at 15:30 comment added Mark Grant This is pure speculation, but can you use the map $U(n)\to SP^n(S^1)$ which takes a unitary matrix to its spectrum of eigenvalues? The symmetric power is not a manifold, but maybe its local homology is easier to relate to multiplicities of eigenvalues than that of $\mathcal{D}$.
S May 4, 2017 at 22:33 history bounty started John Klein
S May 4, 2017 at 22:33 history notice added John Klein Authoritative reference needed
May 3, 2017 at 15:15 comment added Neil Strickland You can filter $U(n)$ by the dimension of the relevant eigenspace. The $k$'th filtration quotient is (essentially by the Cayley transform) the Thom space of the bundle of hermitian endomorphisms of the tautological bundle over the Grassmannian $G_k(\mathbb{C}^n)$. This circle of ideas also shows that $\mathcal{D}^c$ is just $\mathbb{R}^{n^2}$. The filtration is stably split by a theorem of Haynes Miller. This does not obviously answer your question, but it seems like relevant context.
May 3, 2017 at 0:23 history edited John Klein CC BY-SA 3.0
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S May 2, 2017 at 23:03 history edited John Klein CC BY-SA 3.0
Three small typos: deleted one parenthesis, and added another. Added period.
S May 2, 2017 at 23:03 history suggested jeq CC BY-SA 3.0
Three small typos: deleted one parenthesis, and added another. Added period.
May 2, 2017 at 22:48 review Suggested edits
S May 2, 2017 at 23:03
May 2, 2017 at 22:48 history edited John Klein CC BY-SA 3.0
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May 2, 2017 at 22:25 history asked John Klein CC BY-SA 3.0