---- > [!definition] Definition. ([[(homological) orientation of a manifold]]) > Recall that for a $d$-[[manifold]] $M$ and $x \in M$, [[local homology of a manifold|we know that]] $H_{i}(M | x; R) \cong \begin{cases} > R & i =d; \\ > 0 & i \neq d. > \end{cases}$ > A **local $R$-orientation of $M$ at $x$** is a choice of $R$-[[module]] [[submodule generated by a subset|generator]] $\mu_{x} \in H_{d}(M | x; R)$. > > > An **$R$-orientation of $M$** is a collection $\{ \mu_{x} \}_{x \in M}$ of local $R$-orientations such that if $\varphi: \mathbb{R}^{d} \to U \subset M$ > is a [[coordinate patch|patch]] of $M$, and $p,q \in \mathbb{R}^{d}$ with $\varphi(p):=x$ and $\varphi(q):=y$, then the chain of [[isomorphism|identifications]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAAkB9KACgFkABAB8BAHVH0ATmgAWWHmgCUAxSAC+pdJlz5CKAIzkqtRizZdeAVWFiJdaXIWLVGrdjwEiAJiPV6zVkQObh5xAFs6HBkAI2jgACU1AD0oGyUXTRAMd11vUn1jfzMgi1DRCKjYhOTUkQBHDLcdTwN8wtNA4KsbcSlZeQbGrO0PPRI2vw7zEMERXvt+nkGXYxgoAHN4IlAAM0kIMKQfEBwIJEMTALZ5hyxOACpSW2xD6gY6aJgGAAUR3KDJFh1jIcOpMnsDkgAMzUU5IAAskyuQRu-QeT3ELzBu32h0QZBOZ0QFyKnUxWEOrhAELxxzhiBhl2KIHJlPBuKQAFZYUTEUyyaIsW8Pl9fjkWiBAcDQWoKGogA > \begin{tikzcd} > H_d(M | \varphi(p) ) \arrow[r, "\sim"] & H_d(U | \varphi(p)) & H_d(\mathbb{R}^d | p)) \arrow[l, "{\varphi_*, \sim}"'] \arrow[d, "\sim"] \\ > H_d(M | \varphi(q))) \arrow[r, "\sim"'] & H_d(U | \varphi(q)) & H_d(\mathbb{R}^d | q) \arrow[l, "{\varphi_*, \sim}"] > \end{tikzcd} > \end{document} > ``` > > sends $\mu_{x} \mapsto \mu_{y}$.[^1] > [!note] Note. > It is true that an orientation of a manifold is just an orientation of the [[tangent bundle]], but we don't go down that route, for the explicit description given here is useful later on. > > Similar to the case with [[vector bundle|vector bundles]], every [[manifold]] has a unique $\mathbb{F}^{2}$-orientation (there is only one choice of generator). Also similar to the case with vector bundles, the best way to check $R$-orientability is 'using transition maps': [[transitional criterion for manifold (homological) orientability]] ^note > [!basicproperties] > Lots of analogues to the setting of [[orientation of a vector bundle|vector bundle orientations]]: > - Just as with [[vector bundle|vector bundles]], every [[manifold]] has a unique $\mathbb{F}_{2}$-orientation > - The analogue to the [[determinant criterion for orientability of a vector bundle]] is: [[transitional criterion for manifold (homological) orientability]] > - Something even analogous to (the first two parts) of the [[The Thom isomorphism theorem|Thom theorem]] for [[orientation of a vector bundle|oriented]] [[vector bundle|vector bundles]] holds for oriented manifolds... see [[The Thom Theorem for oriented manifolds]]. ^properties [^1]: Here, the unlabeled horizontal isomorphisms are from [[the excision theorem|excision]] (corollary). The vertical isomorphism is [[singular (co)chain map and homomorphism induced by a continuous map||induced by]] [[homotopy invariance of singular homology|translation]] of $\mathbb{R}^{d}$, a [[homeomorphism]]. $\varphi_{*}$ is the map [[relative homology of an embedding of chain complexes|induced on]] [[relative singular homology|relative homology]] by the [[topological pair|map of pairs]] $(\mathbb{R}^{d}, p) \xrightarrow[\sim]{\varphi} (U, \varphi(p))$, a [[homeomorphism]]. ---- #### - [ ] connection e.g. to [[orientation of a smooth manifold]] — a manifold is oriented in that sense iff it is $\mathbb{Z}$-oriented in this one ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```