Related papers: The Samelson Product and Rational Homotopy for Gau…

Let $\mathcal{G}_{k,n}$ be the gauge group of the principal $\mathrm{Sp}(n)$-bundle over $S^4$ corresponding to $k\in\mathbb{Z}\cong\pi_3(\mathrm{Sp}(n))$. We refine the result of Sutherland on the homotopy types of $\mathcal{G}_{k,n}$ and…

Let $M_{l,m}$ be the total space of the $S^3$-bundle over $S^4$ classified by the element $l\sigma+m\rho\in{\pi_4(SO(4))}$, $l,m\in\mathbb Z$. In this paper we study the homotopy theory of gauge groups of principal $G$-bundles over…

We examine the relation between the gauge groups of $\mathrm{SU}(n)$- and $\mathrm{PU}(n)$-bundles over $S^{2i}$, with $2\leq i\leq n$, particularly when $n$ is a prime. As special cases, for $\mathrm{PU}(5)$-bundles over $S^4$, we show…

Given a principal bundle over a closed manifold, G --> P --> M, let P^{Ad} --> M be the associated adjoint bundle. Gruher and Salvatore showed that the Thom spectrum (P^{Ad})^{-TM} is a ring spectrum whose corresponding product in homology…

Let P be a principal bundle with semisimple compact simply connected structure group G over a compact simply connected four-manifold M. In this note we give explicit formulas for the rational homotopy groups and cohomology algebra of the…

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 $k\in\mathbb{Z}$. It is known that the homotopy type of…

The (non)triviality of Samelson products of the inclusions of the spheres into p-regular exceptional Lie groups is completely determined, where a connected Lie group is called p-regular if it has the p-local homotopy type of a product of…

Let $\pi: P\to B$ be a locally trivial fiber bundle over a connected CW complex $B$ with fiber equal to the closed symplectic manifold $(M,\om)$. Then $\pi$ is said to be a symplectic fiber bundle if its structural group is the group of…

In this note, we present a new proof of the isomorphism $\pi_1(SO^+(p,q)) \cong \pi_1(SO(p))\times \pi_1(SO(q))$ using the long exact sequence associated to a fibration. While this formula is already known, the method of proof presented…

The pro-\'etale fundamental group of a scheme, introduced by Bhatt and Scholze, generalizes the usual \'etale fundamental group $\pi_1^{\mathrm{et}}$ defined in SGA1 and leads to an interesting class of "geometric coverings" of schemes,…

Let $m$ and $n$ be two positive integers such that $m < n$. Denote by $P_{n,k}$ the principal $Sp(n)$-bundle over $S^{4m}$ and $\mathcal{G}_{k,m}(Sp(n))$ be the gauge group of $P_{n,k}$ classified by $k\varepsilon'$, where $\varepsilon'$ is…

We study an effective gauge theory whose gauge group is a semidirect product $G = G_c \rtimes \mathit{\Gamma}$ with $G_c$ and $\mathit{\Gamma}$ being a connected Lie group and a finite group, respectively. The semidirect product is defined…

Under certain conditions, we describe the homotopy type of the homo-topy fibre of the inclusion map F\_n(X) $\rightarrow$ $\prod$\_1^n X for the n-th configuration space F\_n(X) of a topological manifold X without boundary such that dim(X)…

Let $Y$ be a pointed space and let $\mathcal E(Y^r)$ be the group of based self-equivalences of $Y^r$, $r\geq 2$. For $Y$ a homotopy commutative $H$-group we construct a subgroup $\mathcal E_{\mathrm{Mat}}(Y^r)$ of $\mathcal E(Y^r)$ which…

If P \to X is a topological principal K-bundle and \hat K a central extension of K by Z, then there is a natural obstruction class \delta_1(P) in \check H^2(X,\uline Z) in sheaf cohomology whose vanishing is equivalent to the existence of a…

A Lie group is called $p$-regular if it has the $p$-local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into $p$-regular $\mathrm{SO}(2n)_{(p)}$ is determined, which…

Let w: Map(X,Y;f) -> Y denote a general evaluation fibration. Working in the setting of rational homotopy theory via differential graded Lie algebras, we identify the long exact sequence induced on rational homotopy groups by w in terms of…

Let $G$ be a compact connected Lie group with $\pi_1(G)\cong\mathbb{Z}$. 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 gauge group of a principal $G$-bundle $P$ over a space $X$ is the group of $G$-equivariant homeomorphisms of $P$ that cover the identity on $X$. We consider the gauge groups of bundles over $S^4$ with $\mathrm{Spin}^c(n)$, the complex…

Let Aut(p) denote the topological monoid of self-fibre-homotopy equivalences of a fibration p:E\to B. We make a general study of this monoid, especially in rational homotopy theory. When E and B are simply connected CW complexes with E…