Let $M$ be a differentiable manifold of dimension $n$. First I give two definitions of Orientability.

The first definition should coincide with what is given in most differential topology text books, for instance Warner's book.

Orientability using differential forms: There exists a nowhere vanishing differential form $\omega$ of degree $n$ on $M$.

The second one is from Greenberg and Harper, "Algebraic Topology". This is the "fundamental class" approach. Let $x$ be a point on $X$, and let $R$ be a commutative ring and in the following the homologies are with coefficients in $R$.

Local orientability: A local $R$ orientation of $X$ at $x$ is a choice of a generator of the $R$-module $H_n(X, X-x)$.

By a simple application of Excision, it is seen that the above homology module is indeed isomorphic to $R$. We can also so arrange a neighborhood around every point that this local orientation can be "continued to a neighborhood" and is "coherent". Forgive me for being imprecise here; the detailed lemmas are in the reference given above. With this background in mind, we define:

A Global $R$-orientation of $X$ consists of: 1. A family $U_i$ of open sets covering of $X$, 2. For each $i$, a local orientation $\alpha_i \in H_n(X, X -U_i)$ of along $X$, such that a "compatibility condition" holds.

Here again I am imprecise about the compatibility condition; please check in the reference given above for details. I mean this basically as a question for those who already know both the definitions, as fully writing down the second definition would take 2-3 pages with all the necessary lemmas.

Also we define "orientation" to be a such a global choice.

Now the question:

How do the two definitions, the first one using differential forms, and the second one using homology, match?

Of course, to match we have to take $\mathbb{Z}$ to be the base ring for homology. A related question is about the meaning of orientability and orientation when we take a base ring other than $\mathbb{Z}$. It is nice when the base ring is $\mathbb{Z}/2\mathbb{Z}$; every manifold is orientable. But what on earth does it mean to have $4$ possible orientations for the circle or real line for instance, when you take the base ring to be $\mathbb{Z}/5\mathbb{Z}$?

Also I ask, are there any additional ways to define orientability/orientation for a differentiable manifold(not just for a vector space)?

If anybody has objection to the formulation of the question, please comment below and I will make it community wiki so that it can be edited by anyone.

Let $M$ be a differentiable manifold of dimension $n$. First I give two definitions of Orientability.

The first definition should coincide with what is given in most differential topology text books, for instance Warner's book.

Orientability using differential forms: There exists a nowhere vanishing differential form $\omega$ of degree $n$ on $M$.

The second one is from Greenberg and Harper, "Algebraic Topology". This is the "fundamental class" approach. Let $x$ be a point on $X$, and let $R$ be a commutative ring and in the following the homologies are with coefficients in $R$.

Local orientability: A local $R$ orientation of $X$ at $x$ is a choice of a generator of the $R$-module $H_n(X, X-x)$.

By a simple application of Excision, it is seen that the above homology module is indeed isomorphic to $R$. We can also so arrange a neighborhood around every point that this local orientation can be "continued to a neighborhood" and is "coherent". Forgive me for being imprecise here; the detailed lemmas are in the reference given above. With this background in mind, we define:

A Global $R$-orientation of $X$ consists of: 1. A family $U_i$ of open sets covering of $X$, 2. For each $i$, a local orientation $\alpha_i \in H_n(X, X -U_i)$ of along $X$, such that a "compatibility condition" holds.

Here again I am imprecise about the compatibility condition; please check in the reference given above for details. I mean this basically as a question for those who already know both the definitions, as fully writing down the second definition would take 2-3 pages with all the necessary lemmas.

Also we define "orientation" to be a such a global choice.

Now the question:

How do the two definitions, the first one using differential forms, and the second one using homology, match?

Of course, to match we have to take $\mathbb{Z}$ to be the base ring for homology. A related question is about the meaning of orientability and orientation when we take a base ring other than $\mathbb{Z}$. It is nice when the base ring is $\mathbb{Z}/2\mathbb{Z}$; every manifold is orientable. But what on earth does it mean to have $4$ possible orientations for the circle or real line for instance, when you take the base ring to be $\mathbb{Z}/5\mathbb{Z}$?

Also I ask, are there any additional ways to define orientability/orientation for a differentiable manifold(not just for a vector space)?

If anybody has objection to the formulation of the question, please comment below and I will make it community wiki so that it can be edited by anyone.

Let $M$ be a differentiable manifold of dimension $n$. First I give two definitions of Orientability.

The first definition should coincide with what is given in most differential topology text books, for instance Warner's book.

Orientability using differential forms. : There exists a nowhere vanishing differential form $\omega$ of degree $n$ on $M$.

The second one is from Greenberg and Harper, "Algebraic Topology". Let $x$ be a point on $X$, and let $R$ be a commutative ring and in the following the homologies are with coefficients in $R$.

Local orientability: A local $R$ orientation of $X$ at $x$ is a choice of a generator of the $R$-module $H_n(X, X-x)$.

By a simple application of Excision, it is seen that the above homology module is indeed isomorphic to $R$. We can also so arrange a neighborhood around every point that this local orientation can be "continued to a neighborhood" and is "coherent". Forgive me for being imprecise here; the detailed lemmas are in the reference given above. With this background in mind, we define:

A Global $R$-orientation of $X$ consists of: 1. A family $U_i$ of open sets covering of $X$, 2. For each $i$, a local orientation $\alpha_i \in H_n(X, X -U_i)$ of along $X$, such that a "compatibility condition" holds.

Here again I am imprecise about the compatibility condition; please check in the reference given above for details. I mean this basically as a question for those who already know both the definitions, as fully writing down the second definition would take 2-3 pages with all the necessary lemmas.

Also we define "orientation" to be a such a global choice.

Now the question:

How do the two definitions, the first one using differential forms, and the second one using homology, match?

Of course, to match we have to take $\mathbb{Z}$ to be the base ring for homology. A related question is about the meaning of orientability and orientation when we take a base ring other than $\mathbb{Z}$. It is nice when the base ring is $\mathbb{Z}/2\mathbb{Z}$; every manifold is orientable. But what on earth does it mean to have $4$ possible orientations for the circle or real line for instance, when you take the base ring to be $\mathbb{Z}/5\mathbb{Z}$?

If anybody has objection to the formulation of the question, please comment below and I will make it community wiki so that it can be edited by anyone.

Let $M$ be a differentiable manifold of dimension $n$. First I give two definitions of Orientability.

The first definition should coincide with what is given in most differential topology text books, for instance Warner's book.

Orientability using differential forms. There exists a nowhere vanishing differential form $\omega$ of degree $n$ on $M$.

The second one is from Greenberg and Harper, "Algebraic Topology". Let $x$ be a point on $X$, and let $R$ be a commutative ring and in the following the homologies are with coefficients in $R$.

Local orientability: A local $R$ orientation of $X$ at $x$ is a choice of a generator of the $R$-module $H_n(X, X-x)$.

By a simple application of Excision, it is seen that the above homology module is indeed isomorphic to $R$. We can also so arrange a neighborhood around every point that this local orientation can be "continued to a neighborhood" and is "coherent". Forgive me for being imprecise here; the detailed lemmas are in the reference given above. With this background in mind, we define:

A Global $R$-orientation of $X$ consists of: 1. A family $U_i$ of open sets covering of $X$, 2. For each $i$, a local orientation $\alpha_i \in H_n(X, X -U_i)$ of along $X$, such that a "compatibility condition" holds.

Here again I am imprecise about the compatibility condition; please check in the reference given above for details. I mean this basically as a question for those who already know both the definitions, as fully writing down the second definition would take 2-3 pages with all the necessary lemmas.

Now the question:

How do the two definitions, the first one using differential forms, and the second one using homology, match?

Of course, to match we have to take $\mathbb{Z}$ to be the base ring for homology.

If anybody has objection to the formulation of the question, please comment below and I will make it community wiki so that it can be edited by anyone.