Given an Ehresmann connection on a fibre bundle, is it possible to define a covariant derivative that measures the rate of change of a section of the fibre bundle as you move through the base manifold? It seems that the usual covariant derivative on the tangent bundle $TM$ of the base manifold $M$ is a […]

The eta invariant $\eta$ of Dirac operator is the generator of the $Pin^+$ bordism group $\Omega_4^{Pin^+}(pt)=\mathbb{Z}_{16}$, see Ref. How two show: 1) For $\mathbb{RP}^4$, we have $16 \eta(\mathbb{RP}^4)=1$, the eta invariant $\eta$ is order 16, associated to the standard Dirac operator. 2) the eta invariant $\eta$ is trivial for its orientation double cover $S^4$. Thus […]

I have read the solution in Vector bundles of rank $k$ with base $S_{1}$, and I am sure that answer is correct. But I have another approach and I don’t know where did I make a mistake. So as circle is covered by real line, and any bundle on real line is trivial as real […]

Definition 1 : Let $B$ be a topological space. A complex vector bundle of rank $n$ over $B$ consists of the data $(E,B,\pi,\{U_i\}_i,\{\phi_i\}_i)$ where $E$ is a topological space, $\pi:E\to B$ is a continuous map called the projection map, $\{U_i\}_i$ is an open cover of $B$ and $\phi_i:\pi^{-1}(U_i)\to U_i\times \mathbb C^n$ is a homeomorphism such […]

As we know, $S^0\to S^n \to \mathbb RP^n$ is a fiber bundle with the usual covering space projection. I wonder if we can construct a projection $S^n \to S^n$ such that $S^0 \to S^n \to S^n$ becomes a fiber bundle. I need this result to complete the proof in here.

Problem (1) Classify all $S^1$ bundles over the base manifold $S^2$. (2) Do the same question for $S^2$ bundles. Moreover, does there exist a universal method to solve this kind of problem? What mathematical tool and concept should be required? And, could someone suggest some reference books related to the problem? (I’m not sure my […]

I just noticed something funky. Let $X$ denote an $I$-indexed family of sets. There is a projection $$\pi_X: \bigsqcup_{i:I} X_i \rightarrow I.$$ It isn’t necessarily surjective, of course, because one or more of the $X_i$ may be empty. Anyway, I noticed that the set $\prod_{i:I} X_i$ can be identified with the set of sections of […]

Real, complex and quaternionic versions of Hopf fibration ($S^0\to S^n\to\mathbb RP^n$, $S^1\to S^{2n+1}\to\mathbb CP^n$ and $S^3\to S^{4n+3}\to\mathbb HP^n$) give rise to spherical fibrations $S^1\to\mathbb RP^{2n+1}\to\mathbb CP^n$ and $S^2\to\mathbb CP^{2n+1}\to\mathbb HP^n$. Question. Does octonionic Hopf fibration $S^7\to S^{15}\to\mathbb OP^1$ give rise to a fibration $\mathbb HP^3\to\mathbb OP^1$? (On one hand, constructing a map $\mathbb HP^3\to\mathbb OP^1$ […]

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that $\phi_U:p^{-1}(U)\longrightarrow U\times \mathbb{K}^n$,$\phi_V:p^{-1}(V)\longrightarrow V\times \mathbb{K}^n$. $(b,x)\in E$. Then $\phi_U(b,x)=(b,u)$, $u\in \mathbb{K}^n$, $\phi_V(b,x)=(b,v)$, $v\in \mathbb{K}^n$. Then $\phi_V\phi_U^{-1}(b,u)=(b,v)$, $v=\lambda(b)u$. Thus for any $(b,x_1\otimes x_2)\in F_b(\xi_1\otimes \xi_2)$, […]

Suppose that $p_1:E_1\to B$, $p_2:E_2 \to B$ are two $C^\infty$ fiber bundles which are $C^1$ isomorphic. That is, there exists a $C^1$ diffeomorphism $f:E_1\to E_2$ satisfying $p_2 \circ f = p_1$. Question: Does it follow that $p_1:E_1\to B$ and $p_2:E_2 \to B$ are $C^\infty$ isomorphic? Motivation: If $M, N$ are two $C^\infty$ manifolds which are […]

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