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$f: X \to Y$ and $Z$ are appropriate for intersection theory $X,Y,Z$ are boundaryless oriented manifolds, $X$ is compact, $Z$ is closed submanifold of $Y$, and $\dim X + \dim Z = \dim Y$), $f$ is transversal to $Z$. According to the text:

The orientation of $X$ provides an orientation of $df_xT_x(X).$ Then the orientation number at $x$ is $+1$ if the orientation on $df_xT_x(X)$ and $T_z(Z)$ "add up" to the prescribed orientation on $Y$. -- Guillemin and Pollack, Differential Topology Page 108

So I am totally confused. What does it mean by "add up" to the prescribed orientation? Does it simply mean first $df_xT_x(X)$ and then $T_z(Z)$ counterclockwisely?

Definition: Orientation of $V$, a finite-dimensional real vector space: Let $\beta, \beta^\prime$ be ordered basis of $V$, then there is a unique linear isomorphism $A: V \to V$ such that $\beta = A \beta^\prime$. The sign given an ordered basis $\beta$ is called its orientation.

Definition: Orientation of $X$, a manifold with boundary: A smooth choice of orientations for all the tangent space $T_x(X).$

1LiterTears
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  • Oh yeah, thanks a lot tomasz. The idea of counterclockwise was a wrong guess from a latter illustration on page 112. Thank you so much for helping me clear this out! – 1LiterTears Jul 21 '13 at 21:15
  • Well, I'll post it as an answer then. :) – tomasz Jul 21 '13 at 21:17

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What does counterclockwisely even mean in this context? You just take a postively oriented basis in $T_x(X)$, transport it by $df$, and then add to it a postively oriented basis of $Z$ and check whether or not they give a positively oriented basis of the tangent space. Or you could do it in the other order, it doesn't really matter, as long as you are consistent, only the sign will change.

tomasz
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  • What is positively oriented basis of the tangent space? Does it have something to do with counterclockwise/right hand rule/pointing outward...? – 1LiterTears Jul 21 '13 at 21:23
  • Not in general. Counterclockwise or clockwise each define an oriention for the plane and left/right hand rules define orientations of the three dimensional space. Here it means just a basis that is positively oriented according to the given orientation on $Z$. – tomasz Jul 21 '13 at 21:45
  • Yeah, but seems there is a canonical orientation, so I was asking if it is the counterclockwise one. – 1LiterTears Jul 21 '13 at 21:47
  • I put all the details in this question. :-) Hope you don't mind taking a look at: http://math.stackexchange.com/questions/449055/plus-or-minus-is-there-a-canonical-orientation-such-like-counterclockwisely – 1LiterTears Jul 21 '13 at 21:53