Mistake in Spivak's definition of a consistent orientation on a manifold

Question

I think Spivak may have made a mistake when defining the notion of a consistent orientation on a manifold. I've included the relevant section of Calculus on Manifolds below. I should mention that when Spivak mentions a manifold, they are referring to an embedded submanifold in $\mathbb{R}^n$.

It is often necessary to choose an orientation $\mu_x$ for each tangent space $M_x$ of a manifold $M$. Such choices are called consistent provided that for every coordinate system $f\colon W\to\mathbb{R}^n$ and $a,b\in W$ the relation $$[f_*((e_1)_a),\ldots,f_*((e_k)_a)]=\mu_{f(a)}$$ holds if and only if $$[f_*((e_1)_b),\ldots,f_*((e_k)_b)]=\mu_{f(b)}.$$

I think this definition of consistent orientation is problematic when $W$ is not a connected set. For instance, consider $$M=\{(x,y)\in\mathbb{R}^2:y=0\}.$$ Then $M$ is a $1$-manifold in $\mathbb{R}^2$. However under Spivak's definitions, $M$ is non-orientable. Indeed, suppose for the sake of contradiction that $\mu$ is a consistent orientation on $M$. Define $W=(-1,1)\cup (2,4)\subset\mathbb{R}$ and the coordinate systems $f,g\colon W\to\mathbb{R}^2$ by $$f(x)=(x,0)\qquad\text{and}\qquad g(x)=\begin{cases}(-x,0), &\text{if }x\in(-1,1);\\(x,0),&\text{if }x\in(2,4).\end{cases}$$ Finally, set $a=0$ and $b=3$. Then $f(a)=g(a)=(0,0)$ and $f(b)=g(b)=(3,0)$.

According to Spivak's definition, we must have $[f_*((e_1)_a)]=\mu_{(0,0)}$ if and only if $[f_*((e_1)_b)]=\mu_{(3,0)}$. Similarly, $[g_*((e_1)_a)]=\mu_{(0,0)}$ if and only if $[g_*((e_1)_b)]=\mu_{(3,0)}$. This is impossible, since $$[f_*((e_1)_a)]=-[g_*((e_1)_a)]\qquad\text{but}\qquad [f_*((e_1)_b)]=[g_*((e_1)_b)].$$ I think a similar construction shows that every manifold is non-orientable with Spivak's definition. How do we fix this definition? Does it suffice to insist that $W$ be connected?

Answer 1

You are right, Spivak is imprecise. In Theorem 5.2 he introduces the concept of a coordinate system around $x \in M$ as $1$-$1$ differentiable function $f: W \to \mathbb R^n$ with suitable properties, but he does not require that $W \subset \mathbb R^k$ is connected. As Ted Shifrin remarked, this is absolutely standard.

As you correctly explained, Spivak's definition does not work in the given form. You quoted the definition in your question. The next sentences in Spivak are

Suppose orientations $\mu_x$ have been chosen consistently. If $f: W \to \mathbb R^n $ is a coordinate system such that $$[f_*((e_1)_a),\ldots,f_*((e_k)_a)]=\mu_{f(a)}$$ for one, and hence for every $a \in W$, then $f$ is called orientation-preserving. If $f$ is not orientation-preserving and $T: \mathbb R^k \to \mathbb R^k$ is a linear transformation with $\det T = -1$, then $f \circ T$ is orientation-preserving. Therefore there is an orientation- preserving coordinate system around each point.

The assertion concerning a not orientation-preserving $f$ is definitely wrong. It is true provided $W$ is connected. For non-connected $W$ there are coordinate systems $f$ such that neither $f$ nor $f \circ T$ are orientation-preserving. However, the last sentence "Therefore there is an orientation-preserving coordinate system around each point" is true.

So what is the correct definition? Call a coordinate system $f$ orientation-preserving if for all $b \in W$ $$[f_*((e_1)_b),\ldots,f_*((e_k)_b)]=\mu_{f(b)}.$$ Then the family $\mu_x$ is called consistent if there is an orientation-preserving coordinate system around each point. One can easily prove that $\mu_x$ is consistent if and only if for each that for every coordinate system $f\colon W\to\mathbb{R}^n$ with a connected $W$ and all $a,b\in W$ the relation $$[f_*((e_1)_a),\ldots,f_*((e_k)_a)]=\mu_{f(a)}$$ holds if and only if $$[f_*((e_1)_b),\ldots,f_*((e_k)_b)]=\mu_{f(b)}.$$