# Questions tagged [orientation]

For question regarding the notion of orientation both in topology and in global analysis.

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### Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $[M \mathbin\sharp N] \in H_n(M \mathbin\sharp N)$ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$[M \mathbin\sharp N] = [M]+[N]$$ which ...
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### What would be the concept of “Infinite Dimension Orientability”?

We have the following definition of orientability for finite dimensional vector spaces: Definition: Let $\Bbb{E}$ be a finite dimension vector space and $\mathscr{B}(\Bbb{E})$ be the set of all ...
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### A vector bundle which has an orientation-reversing isomorphism has a subbundle of rank $1$?

Let $E$ be a smooth real vector bundle of even rank, over a smooth manifold $M$. Suppose there exist an orientation-reversing vector bundle isomorphism $\Phi:E \to E$. Is it true that $E$ has a ...
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### How to define orientation on infinite dimensional vector space

Let $\mathbb{V}$ be a real Banach space (if someone knows the answer for more arbitrary T.V.S. then great). Is there some concept of orientation that one could define on $\mathbb{V}$ that matches the ...
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### Orientability of variety

Consider the real projective variety on $\mathbb{R}P^n \times \mathbb{R}P^n$ defined by the bihomogeneous polynomial $x_0y_0+...+x_ny_n$, i.e. $V=\{([x_0:...:x_n],[y_0:...:y_n])|x_0y_0+...+x_ny_n=0\}$....
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### Induced orientation on manifold

Suppose $M$ is a naturally oriented $n$-manifold embedded in $\mathbb{R}^n$ with non-empty boundary. Give $\partial M$ the induced orientation. The induced orientation defines a unit normal vector ...
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### Stokes' theorem: Induced orientation on the boundary of a manifold

The Question Let $K = \{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 \geq 1\}$, where $K$ is oriented via the canonical volume form on $\mathbb{R}^3$: $dx \wedge dy \wedge dz$. Let $\mathbb{S}^2$ be ...
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### Show that the boundary orientation agrees with the standard orientation of $\mathbb R^{k-1}$ if and only if k is even.

$H^{k}$ is oriented by the standard orientation of $\mathbb R^{k}$. Thus $\partial$$H^{k} acquires a boundary orientation. But \partial$$H^{k}$ may be identified with $\mathbb R^{k-1}$. Show that ...
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### Interpretations of orientability on higher order homology groups

A surface $S$ is orientable iff its first homology group $H_1(S)$ is a free abelian group $F$, or it is non-orientable iff $H_1(S)$ has the form $F + \mathbb{Z}_2$ (so says wiki, "orientability and ...