For a torsion free sheaf $E$ on a surface $X$ we have $0\rightarrow E\rightarrow E^{**}\rightarrow Z\rightarrow 0$ where $Z$ is a zero dimensional subscheme of $X$. Let $p\in\mathbb{P}^2$ be a closed point. Let $\mathcal{I}_p$ be the corresponding ideal sheaf. We have $(\mathcal{I}_p^{\oplus 2})^{**}=\mathcal{O}^{\oplus 2}$. Hence $\mathcal{O}^{\oplus 2}/\mathcal{I}_p^{\oplus 2}$ is a zero dimensional subscheme of $\mathbb{P}^2$. […]

I’m trying to construct an explicit isomorphism from $E = \{([x], v) : [x] ∈ \Bbb{R}P^1, v ∈ [x]\}$ to $T = [0, 1] × R/ ∼$ where $(0, t) ∼ (1, −t)$. I verified that $\Bbb{R}P^1$ is homeomorphic to $\Bbb{S}^1$ which is homeomorphic to $[0,1]/∼$ where $0∼1$. So this is the map I have […]

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 […]

Suppose that $L \to M$ is complex line bundle over a manifold $M$. One can therefore form the dual bundle $L^* \to M$. We can identify $L^* \otimes L$ with endomorphism bundle $End(L)$. Why it is true that $End(L)$ is trivial? Why the assumption of being line bundles is essential? Is it also true for […]

In my notes, the lecturer considers a smooth vector field $v: TM\to T(TM)$, with $M$ a smooth manifold. Let’s write $$v(u,e)=((u,e), (a(u,e),b(u,e)).$$ It is said that $v$ is a second order equation if $T\pi_M\circ v=\text{id}$. It implies that $a(u,e)=e$, i.e. $$v(u,e)=((u,e),(e,b(u,e))).$$ Here, my notes claim that if $c(t)=(u(t),e(t))$ is a curve on $TM$ satisfying the […]

Stiefel-Whitney classes are defined (for example, in Milnor’s Characteristic classes) as elements of the cohomology groups with $\mathbb{Z}_2$-coefficients as follows. $$w_i(E)=\phi^{-1} Sq^i(u)$$ where $E$ is an $n$-plane bundle over $B$, $E_0$ is the complement of the zero section, $Sq^i: H^*(E,E_0)\to H^{*+i}(E,E_0)$, $u\in H^n(E,E_0)$ is the Thom class and $\phi: H^*(B)\to H^{*+n}(E,E_0)$ is the Thom isomorphism […]

My last question (Is such a map always null-homotopic?) is quite similar. If you do not care about my motivation for these questions, you can skip to the last line. I asked if some assumptions were sufficient conditions to guarantee the non-existence of “phantoms” (essential maps that are non-essential on every finite sub-complex). Now I […]

Suppose that $X$ is a compact (smooth) $n$-manifold and $E \to X$ be a rank $N$ smooth (complex) vector bundle. Choose finite covering $(U_i)_i$ by domains of the charts $\varphi:U_i \to V_i \subset \mathbb{R}^n$ and let $h_i:E|_{U_i} \to V_i \times \mathbb{C}^N$ be a collection of trivialisations (composed with $\varphi_i \times id$ in order to land […]

Let $A, B, C$ be holomorphic vector bundles over some complex manifold $X$. Let $A’, B’, C’$ be sub bundles, respectively. Suppose that we have short exact sequences: $$0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ $$0 \rightarrow A’ \rightarrow B’ \rightarrow C’ \rightarrow 0$$ Let’s also suppose that the quotient sequence $$0 \rightarrow […]

I’m trying to study line bundle over $S^2$. In this post was outlined the method based on clutching functions. But now I’m interesting in another approach. For the sphere there is two maps : upper hemisphere and lower hemisphere with intersection as $[-\epsilon,\epsilon]\times S^1$. For the upper hemisphere and lower hemisphere its well-known that bundles […]

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