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I'm currently trying to understand the Birman-Williams Template Theorem, proved in the paper "Knotted periodic orbits II: Fibered knots, Low Dimensional Topology". Unfortunately, there doesn't seem to be a free online version but there are two main ways of arguing the statement which I'll reproduce here.

Recall that an invariant set $\Lambda$ is hyperbolic with respect to flow $\phi_t$ if the tangent space restricted to $\Lambda$ can be decomposed into the unstable, stable and center bundle, i.e. $TM_{\Lambda}=E^s\oplus E^u \oplus E^c$. We assume that $M$ is a differentiable manifold of dimension 3 here. By Smale's Decomposition Theorem, we can decompose a special kind of hyperbolic invariant set $\Lambda$ known as the chain-recurrent set into finite disjoint closed subsets that were themselves invariant under $\phi_t$, i.e. $\Lambda=B_1\cup B_2 \cup ... \cup B_n$. We call the subsets $B_i$ basic sets.

In the Birman-Williams paper, they argue:

Suppose $B$ is a basic set for $\phi_t$ of dimension 2. It follows that $B$ is an attractor or repeller, as one has $u+s=2$ where $u$ (resp. $s$) is the unstable (resp. stable) dimension.

This is all they say. I don't follow this reasoning - it seems quite mysterious to me.

There's another explanation of this argument in Ghrist et. al.'s monograph on Knots and Links in 3-Dimensional Flows, where they argue:

Since $B$ is two dimensional and hyperbolic and $M$ three-dimensional, the stable, unstable, and center bundles must each be of dimension one. Since $B$ must contain the center bundle, it must also contain either the stable or unstable bundle, leaving only the remaining direction. Hence, $B$ is either an attractor or a repeller.

I don't understand why the stable, unstable and center bundles must each be of dimension one - I accept that the center bundle bundle must be of dimension one, but I don't quite get the rest. It's also not clear to me that $TM_B$ has to be of dimension 2. Intuitively, we know that the tangent bundle of an $n$-dimensional manifold has dimension 2n - while it's true that $B$ need not be a manifold, why shouldn't the same thing be true here as well?

I think the kernel of my confusion resides in the fact that I am unsure about how the notion of "dimension" plays out here. I sometimes wonder if they mean something straightforward like topological dimension, or if they mean something like fractal dimension, but that just seems to confuse things even more...

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  • $\begingroup$ The dimension of bundle here refers only to the tangent space dimension (i.e, it does not include the dimension of base space). $\endgroup$ Sep 1, 2017 at 4:50
  • $\begingroup$ So is the reasoning something like: If $B$ has dimension 2, then the tangent bundle $TM_B$ has dimension 4, including the dimension of $B$ and the tangent space of $B$. And this implies that the tangent space has dimension 2, and since it needs to contain the center bundle (which is of dimension 1), it must contain either the stable or unstable bundle? But how do we know that the unstable and stable bundle must be each of dimension one? $\endgroup$
    – asldjk
    Sep 1, 2017 at 10:23
  • $\begingroup$ Because if unstable manifold U is of dimension 2, then the "1D submanifold" of U in B ( union with 1D center manifold) by itself is not invariant. Same argument if stable manifold is of dimension 2. We need B to be invariant by definition. $\endgroup$ Sep 1, 2017 at 11:53
  • $\begingroup$ So just to clarify, when people say that hyperbolicity requires a decomposition of the tangent bundle $TM_B$ into $E^s \oplus E^u \oplus E^c$, they don't really mean the decomposition of the actual tangent bundle itself but the tangent space at every point of $B$ , right? (Otherwise, I have a hard time imagining how the base space part of the tangent bundle can be decomposed into stable/unstable bundles). And we know that for a 2-dim subspace X of a manifold M, the tangent space of $X$, ie $TM_X$ is itself also of two-dimensions (at every point $x\in X$), even if X is not a manifold, yes? $\endgroup$
    – asldjk
    Sep 1, 2017 at 23:59

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