dc.contributor.advisor | Kronheimer, Peter | en_US |
dc.contributor.author | Xie, Yi | en_US |
dc.date.accessioned | 2017-07-25T14:42:19Z | |
dc.date.created | 2016-05 | en_US |
dc.date.issued | 2016-05-11 | en_US |
dc.date.submitted | 2016 | en_US |
dc.identifier.citation | Xie, Yi. 2016. On the Framed Singular Instanton Floer Homology From Higher Rank Bundles. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. | en_US |
dc.identifier.uri | http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493482 | |
dc.description.abstract | In this thesis we study the framed singular instanton Floer homology defined by by Kronheimer and Mrowka in \cite{KM3}.
Given a 3-manifold $Y$ with a link $K$ and $\delta \in H^2(Y,\mathbb{Z})$ satisfying a non-integral condition, they define the singular instanton Floer homology group $I^N(Y,K,\delta)$ by counting singular flat $PSU(N)$-connections with fixed holonomy around $K$.
Take a point $x\in Y\backslash K$, classical point class operators $\mu_i (x)$ of degree $2i$ on $I^N(Y,K,\delta)$ can be defined as in the original Floer theory defined by smooth connections.
In the singular instanton Floer homology group $I_\ast^N(Y,K,\delta)$, there is a special degree 2 operator $\mu (\sigma)$ for $\sigma \in K$. We study this new operator and obtain a universal relation between this operator and the point class operators $\mu_i (x)$.
After restricted to the reduced framed Floer homology $F\bar{I}_\ast^N(Y,K)$, these point classes operators $\mu_i (x)$ become constant numbers related to the $PSU(N)$-Donaldson invariants of four-torus $T^4$.
Then the universal relation becomes a characteristic polynomial for the operator $\mu(\sigma)$ so that we can understand the eigenvalues of $\mu(\sigma)$ and decompose the Floer homology as eigenspaces. | en_US |
dc.description.sponsorship | Mathematics | en_US |
dc.format.mimetype | application/pdf | en_US |
dc.language.iso | en | en_US |
dash.license | LAA | en_US |
dc.subject | Mathematics | en_US |
dc.title | On the Framed Singular Instanton Floer Homology From Higher Rank Bundles | en_US |
dc.type | Thesis or Dissertation | en_US |
dash.depositing.author | Xie, Yi | en_US |
dc.date.available | 2017-07-25T14:42:19Z | |
thesis.degree.date | 2016 | en_US |
thesis.degree.grantor | Graduate School of Arts & Sciences | en_US |
thesis.degree.level | Doctoral | en_US |
thesis.degree.name | Doctor of Philosophy | en_US |
dc.contributor.committeeMember | Taubes, Clifford | en_US |
dc.contributor.committeeMember | Mrowka, Tomasz | en_US |
dc.type.material | text | en_US |
thesis.degree.department | Mathematics | en_US |
dash.identifier.vireo | http://etds.lib.harvard.edu/gsas/admin/view/867 | en_US |
dc.description.keywords | Floer homology | en_US |
dash.author.email | aleph.xie@gmail.com | en_US |
dash.contributor.affiliated | Xie, Yi | |