Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFiniteness of $A_n$-equivalence types of gauge groups
228
0
0.0
(
0
)
Authors
Mitsunobu Tsutaya
Ask ChatGPT about the research
No Arabic abstract
Let $B$ be a finite CW complex and $G$ a compact connected Lie group. We show that the number of gauge groups of principal $G$-bundles over $B$ is finite up to $A_n$-equivalence for $n<infty$. As an example, we give a lower bound of the number of $A_n$-equivalence types of gauge groups of principal $SU(2)$-bundles over $S^4$.
rate research
Read More
We construct the homotopy pullback of $A_n$-spaces and show some universal property of it. As the first application, we review the Zabrodskys result which states that for each prime $p$, there is a finite CW complex which admits an $A_{p-1}$-form but no $A_p$-form. As the second application, we investigate $A_n$-types of gauge groups. In particular, we give a new result on $A_n$-types of the gauge groups of principal $mathrm{SU}(2)$-bundles over $S^4$, which is a complete classification when they are localized away from 2.
Let $G$ be a compact connected Lie group and let $P$ be a principal $G$-bundle over $K$. The gauge group of $P$ is the topological group of automorphisms of $P$. For fixed $G$ and $K$, consider all principal $G$-bundles $P$ over $K$. It is proved by Crabb--Sutherland and the second author that the number of $A_n$-types of the gauge groups of $P$ is finite if $n<infty$ and $K$ is a finite complex. We show that the number of $A_infty$-types of the gauge groups of $P$ is infinite if $K$ is a sphere and there are infinitely many $P$.
Let $G$ be a compact connected Lie group with $pi_1(G)congmathbb{Z}$. We study the homotopy types of gauge groups of principal $G$-bundles over Riemann surfaces. This can be applied to an explicit computation of the homotopy groups of the moduli spaces of stable vector bundles over Riemann surfaces.
The $p$-local homotopy types of gauge groups of principal $G$-bundles over $S^4$ are classified when $G$ is a compact connected exceptional Lie group without $p$-torsion in homology except for $(G,p)=(mathrm{E}_7,5)$.
Let $M$ be an orientable, simply-connected, closed, non-spin 4-manifold and let $mathcal{G}_k(M)$ be the gauge group of the principal $G$-bundle over $M$ with second Chern class $kinmathbb{Z}$. It is known that the homotopy type of $mathcal{G}_k(M)$ is determined by the homotopy type of $mathcal{G}_k(mathbb{CP}^2)$. In this paper we investigate properties of $mathcal{G}_k(mathbb{CP}^2)$ when $G = SU(n)$ that partly classify the homotopy types of the gauge groups.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
