But the punctured torus \({\mathbb T}^2 \setminus \{pt\}\) can be *immersed* in \({\mathbb R}^2\). In fact, the punctured torus is just a disc with two handles attached, and this can be immersed in the plane if we allow the two handles to cross over one another (the crossing explains why we don’t get an immersion).

In the set-up of the Handle Smoothing Theorem, \({\mathbb R}^2 \) is equipped with a smooth structure which is pulled back via a topological embedding into a smooth surface \(S\). This smooth structure, call it \(\mathcal S\), can then be pulled back to the punctured torus. Let \(T’\) denote the punctured torus equipped with the pulled-back coarse structure.

At this point Hatcher quotes two facts from the theory of smooth surfaces. (In higher dimensional applications of the torus trick, this is where we would appeal to non-simply-connected smooth or PL surgery.)

**Fact 3** The smooth structure \(T’\) on the punctured torus extends to a smooth structure on the unpunctured torus; call \(T\) the unpunctured torus equipped with this smooth structure.

**Fact 4** The (potentially) “exotic” torus \(T\) is in fact diffeomorphic to the standard torus \({\mathbb T}^2\).

Accepting these for now, let us write \(R\) for the universal cover of \(T\), a potentially exotic \({\mathbb R}^2\). Notice that a small open set around 0 in \(R\) is identified (smoothly) with a small open set in the copy of \({\mathbb R}^2\) with which we started the discussion. Let \(\phi\) be a diffeomorphism from \(T\) to \({\mathbb T}^2\). We can write

\[ \phi = \phi_1 \circ \phi_2 \]

where \(\phi_2\) is a diffeomorphism \(T\to {\mathbb T}^2 \) acting trivially on the fundamental group and \(\phi_1\) is given by an element of \(GL(2,{\mathbb Z}) \). If we lift \(\phi_2\) to a periodic diffeomorphism of universal covers then it preserves the integer lattice and thus is *bounded*. This is Fact 2 – and though it is applied to a different copy of \({\mathbb R}^2\) than the one we originally started from, we can restrict attention to a small open set where the two copies agree and that is sufficient.

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A very nice preprint by Hatcher (see below) helps demystify the torus trick by applying it to a *low*-dimensional problem. The theorem in question is that every (closed, for simplicity) topological surface can be given a compatible smooth structure. (As usual the theorem comes along with various relative versions and uniqueness (up to isotopy) to accompany the existence statement.) The use of the torus trick means that the argument only depends on the *differential* topology of the plane and not on difficult \(C^0\) facts like the topological Schoenflies theorem (which is needed in the classical approach to the smoothability of surfaces).

Hatcher deduces the existence of smooth structures from the following

**Handle Smoothing Theorem** Let \(S\) be a smooth surface and \(p\in\{0,1,2\}\). Any topological embedding of \(D^p\times {\mathbb R}^{2-p}\) into \(S\) which is smooth in a neighborhood of \(\partial D^p \times {\mathbb R}^{2-p}\) can be isotoped to a smooth embedding in a neighborhood of \(D^p \times \{0\}\), staying fixed near \(\partial D^p \times {\mathbb R}^{2-p}\) and outside a larger neighborhood of \(D^p\times \{0\}\).

(Note that the final clause is vacuous when \(p=2\).) Here is a sketch of the argument leading from this theorem to the existence of smooth structures: Cover the surface\(S\) by finitely many domains of coordinate charts and inductively extend the smooth structure one chart at a time. To do the inductive step, one needs to know that if \(W\subseteq S\) is an open subset which has already been smoothed, and \(U\subseteq S\) is the domain of a coordinate chart \(h\colon U\to {\mathbb R}^2\), then the embedding

\[k_0= h^{-1}\colon {\mathbb R}^2\to S \]

can be changed by an isotopy \(k_t\) that remains constant outside \(h(W)\) in such a way that the new embedding \(k_1\) is *smooth* on \(h(W)\). To construct such an isotopy, triangulate the open set \(h(W)\subseteq{\mathbb R}^2\) by simplices whose size tends to zero at the boundary. Then apply the handle smoothing theorem first to small balls around all the 0-simplices, then to neighborhoods of the 1-simplices, and finally to what’s left. The “small simplex” condition ensures that the resultant isotopies extend (by the identity) on the rest of \({\mathbb R}^2\).

So how is the handle smoothing theorem to be proved? Hatcher gives separate arguments for the three cases, but the only one that involves the torus trick is the 0-handle case (\(p=0\)), so that’s what I’ll concentrate on here. The basic structure of the argument is the following.

Look at the topological embedding \(h\colon {\mathbb R}^2 \to S\), and use it to pull back the smooth structure of \(S\) to some smooth structure on \({\mathbb R}^2\); let \(X\) denote \({\mathbb R}^2\) equipped with this (potentially exotic) smooth structure. The identity map \({\mathbb R}^2\to X\) is then a (potentially wild) homeomorphism.

**Fact 1** There are no exotic smooth structures on \({\mathbb R}^2\).

Thus the identity map \({\mathbb R}^2\to X\) becomes identified with a homeomorphism from \({\mathbb R}^2\) to itself.

**Fact 2** Any self-homeomorphism \(g\) of \({\mathbb R}^2\) can be written as a composite \( g_1\circ g_2\), where \(g_1\) is a diffeomorphism and \(g_2\) is a *bounded* homeomorphism, i.e., \( |g(x)-x|\) is uniformly bounded.

When \({\mathbb R}^2\) is identified with the interior of a 2-disk, any bounded homeomorphism extends continuously (as the identity) to the boundary and exterior of the disk. Using this trick, it is easy to deduce the 0-Handle Smoothing Theorem from Facts 1 and 2.

The torus trick is used to prove these facts (actually, to prove slightly weaker versions of them, but still sufficient for what is needed). I will explain in the followup post.

Hatcher, Allen. “The Kirby Torus Trick for Surfaces.” *arXiv:1312.3518 [math]* (December 12, 2013). http://arxiv.org/abs/1312.3518.

Kirby, Robion C, and Laurence C Siebenmann. *Foundational Essays on Topological Manifolds, Smoothings, and Triangulations*. Princeton, N.J.: Princeton University Press, 1977.

However a little research led me to the free utility pdfSAM, which “splits and merges” (SAM) pdf files. In particular, it can split up a PDF file into sub-files according to the PDF bookmarks in the original file – and these, in turn, can be auto-generated from the book manuscript using the package “hyperref”.

The upshot of all of this is that I can supply this link to a directory that contains individual files for all the sections of the book – and I can update these in a seamless way.

Of course this playing around is maybe a substitute for actual *writing*, but it still feels like (some sort of) progress.

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So the historical introduction goes: classification of 2-manifolds; Poincaré and 3-manifolds, and the history of the Poincare conjecture; the Fifth Complement to *Analysis Situs* and the Poincaré homology sphere.

Poincaré’s discussion is phrased in terms of (what became known as) a Heegard splitting, but without too much difficulty this can be related to the calculations (using van Kampen’s theorem) to compute the fundamental group below…

We start with 2-disk bundles over the 2-sphere. These are classified by an Euler number. We take 8 copies of this bundle and plumb them together according to the \(E_8\) Dynkin diagram:

Each node of the Dynkin diagram represents a copy of our bundle, and two bundles are plumbed together if the two nodes are linked by an edge. The result (after rounding the corners) is a smooth 4-manifold with boundary; call the boundary \(M\), a closed 3-manifold.

The homology of \(M\) is computed from a complex whose only non-trivial differential is the \(E_8\) matrix. Because this matrix is unimodular, \(M\) is a homology sphere. This part of the computation is the same as in higher dimensions.

However, \(M\) is *not* simply connected. This can be seen as follows. Using van Kampen’s theorem, one computes that the fundamental group of \(M\) has one generator for each node of the Dynkin diagram, and also one relator for each node. Let \(x_k\) be the generator corresponding to node \(k\). The relator corresponding to node \(k\) is of the form

\[ x_k^2 x_{j_1}x_{j_2}\ldots = 1 , \]

where \(j_1,j_2,\ldots \) are the nodes adjacent to \(k\) in the Dynkin diagram. In the case of the diagram above this yields

\[ 1 = x_1^2x_2 = x_2^2x_1x_3 = x_3^2x_4x_5x_2 = x_4^2x_3 = x_5^2x_3x_6 = x_6^2x_5x_7=x_7^2x_8x_6 = x_8^2x_7. \]

Writing \(x=x_1\), \(y=x_8\) allows us to present the group \(\pi_1(M)\) as

\[ \langle x,y | x^3 = y^5 = (xy)^2 \rangle. \]

The permutations \( x = (531) \), \(y = (12345) \) generate the alternating group \(A_5\) and satisfy the above relations (with \(x^3 = y^5 = (xy)^2 = 1\).) Thus \(\pi_1(M) \) surjects onto \(A_5\) and in particular is not trivial. (In fact, it is known that the kernel is of order 2; \(\pi_1(M) \) is the *binary icosahedral group* of order 120.

Thus \(M\) is a homology sphere but not a homotopy sphere.

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- The high-dimensional Poincaré conjecture and the
*h*-cobordism theorem - Milnor’s exotic spheres
- Variation of Pontrjagin classes within a homotopy type

In the version as written I started with Milnor’s examples. That’s because I wanted to get the reader to a point where Milnor has recorded that he arrived in the middle 1950s; he had a smooth, 7-dimensional homotopy sphere, but he didn’t know whether his example was an exotic smooth structure on \(S^7\), or a counterexample to the Poincaré conjecture in dimension 7. But looking at this again, I’ve come to feel that that makes the exposition a bit hard to follow. So I’d like to move the Poincaré discussion earlier (perhaps even before characteristic classes) and then pick up Milnor’s examples, even though this reverses the historical order of Milnor and Smale.

Now the Poincaré discussion has to begin with the original *Analysis Situs* papers and the 3-dimensional conjecture, even if dimension 3 is too low for surgery-type methods to be applicable. I’m working on the exposition here but I’d just like to link to this very beautiful talk by John Morgan at the Clay Institute Conference in 2010:

http://www.youtube.com/watch?v=Utf-uwArrq0

Here are some additional references:

Kirby, R. C., and M. G. Scharlemann. 1979. “Eight faces of the Poincaré homology 3-sphere.” Pp. 113–146 in *Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977)*. New York: Academic Press Retrieved August 29, 2013 (http://www.ams.org/mathscinet-getitem?mr=537730).

Milnor, John. 1956. “On Manifolds Homeomorphic to the 7-Sphere.” *Annals of Mathematics* 64(2):399–405. Retrieved March 5, 2013.

Poincaré, Henri. 2010. *Papers on Topology: Analysis Situs and Its Five Supplements*. American Mathematical Soc. (translated by John Stillwell)

Smale, S. 1990. “The story of the higher dimensional Poincaré conjecture: what actually happened on the beaches of Rio.” *Mathematical Intelligencer* 12:44–51.

Stillwell, John. 2012. “Poincaré and the early history of 3-manifolds.” *American Mathematical Society. Bulletin. New Series* 49(4):555–576. Retrieved August 29, 2013.

Anyhow, here is the next section of the book, which briefly introduces cobordism and the signature theorem, and calculates the first two L-polynomials by hand. I think that for the amateur (i.e. me again) it is important to see first that what’s going on here is a simple “calculate enough examples to fill in the unknown constants”, before one introduces the L polynomial itself, which can seem just like an *ad hoc* mystery. This is a bit like the heat equation proof of the index theorem: it is good, I think, to hack it out by hand for the \(\bar{\partial}\) operator on Riemann surfaces, before getting engaged with the comparing-coefficients calculations in Atiyah, Bott and Patodi. Of course, in that case we now have the Getzler argument also, which yields the correct polynomial directly. I don’t know whether there is something in the same spirit for the signature theorem?

Atiyah, M., R. Bott, and V. K. Patodi. “On the Heat Equation and the Index Theorem.” *Inventiones Mathematicae* 19, no. 4 (December 1, 1973): 279–330. doi:10.1007/BF01425417.

Getzler, E. “Pseudodifferential Operators on Supermanifolds and the Atiyah-Singer Index Theorem.” *Communications on Mathematical Physics* 92 (1983): 163–178.

The basic point about Steenrod squares is to quantify the commutativity of the cup-product operation on a space \(X\). What does this mean? We know that, given cocycles \(a,b\) for \(X\), one has \([ a \smile b] = [b\smile a] \in H^*(X;{\mathbb Z}_2)\). But this doesn’t tell us that the cup-product is commutative “on the nose”: only up to some kind of homotopy. In the chapter as written, I carry this idea forward algebraically, using chain maps and diagonal approximations and so on; as well as defining the Steenrod squares this yields the *symmetric* and *quadratic* constructions that are so important in Ranicki’s algebraic surgery theory. Here though let me sketch how the idea can be implemented *geometrically*, following some lecture notes of Mike Hopkins (see also the very last section of Hatcher’s algebraic topology book).

Let \(X\) be a space and \(x \in H^n(X) \) be a cohomology class (let’s agree that all cohomology groups have mod 2 coefficients). The external square \(x \times x \in H^{2n}(X\times X) \) is represented by a map

\[ X\times X \to K({\mathbb Z}_2,2n) \]

to an Eilenberg-MacLane space. The commutativity of the cup-product gives us a homotopy between this map and its composite with the “flip” automorphism of \(X\times X\). The homotopy can be encoded as a map

\[ (X\times X)\times_{{\mathbb Z}_2} S^1 \to K({\mathbb Z}_2,2n). \]

There are also “higher homotopies”, and taking all of those into account one can promote the above to a map

\[ (X\times X)\times_{{\mathbb Z}_2} S^\infty \to K({\mathbb Z}_2,2n). \]

Notice that \(S^\infty = E{\mathbb Z}_2\), so the left side here is the Borel construction or *homotopy quotient* \( X \times_{h{\mathbb Z}_2} X \) of \(X\times X\) by the involution. The construction shows that the original external square map \( X\times X \to K({\mathbb Z}_2,2n) \) factors through this homotopy quotient.

Okay, so now let’s consider the diagonal map \(\Delta \colon X\to X\times X\). This induces a diagonal map from \(X \times B{\mathbb Z}_2\) to the homotopy quotient. We end up with a diagram of maps like this

\[ \begin{array}{ccccc} H^n(X) & \to & H^{2n}(X\times_{h{\mathbb Z}_2} X) & \to &H^{2n}(X\times X) \\&&\downarrow&&\downarrow\\ && H^{2n}(X \times B{\mathbb Z}_2)&\to & H^{2n}(X) \end{array} \]

where the vertical maps are induced by \(\Delta\). By the Kunneth formula, \( H^*(X \times B{\mathbb Z}_2) \cong H^*(X)[\alpha] \), where \(\alpha\) is the 1-dimensional generator of the cohomology of \( B{\mathbb Z}_2 = {\mathbb{RP}}^\infty \). Thus the induced map (from the diagram above) \( P\colon H^n(X) \to H^{2n}(X \times B{\mathbb Z}_2) \) can be written in components as

\[ P(x) = \sum_{j=0}^n {Sq}^{n-j}(x) \alpha^j, \quad {Sq}^k(x) \in H^{n+k}(X). \]

By construction, \({Sq}^n(x) = x^2. \) The other components of the sum, which measure the extent to which the homotopy commutativity of the cup-product departs from a genuine commutativity, are the *Steenrod squares*.

Hopkins, Michael. “Notes on Steenrod Operations,” 2007. http://isites.harvard.edu/icb/icb.do?keyword=k54584&pageid=icb.page237558

Chapter 3 of Hatcher’s *Vector bundles and K-theory* gives a nice presentation of characteristic class theory. The most current draft version of that book is available from Hatcher’s website. In particular, this contains a fairly detailed presentation of the 2-torsion aspect of real characteristic classes (i.e. Stiefel-Whitney classes), which of course we are going to need later in the book. One learns from the exercises, for instance, the exact formula for the odd Chern classes of the complexified bundle in terms of the Stiefel-Whitney classes, namely

\[ c_{2k+1}(E\otimes{\mathbb C}) = \beta (w_{2k}(E)w_{2k+1}(E) ), \]

where \(\beta\) is the Bockstein homomorphism associated to the exact sequence of coefficient groups

\[ 0 \to {\mathbb Z} \to {\mathbb Z} \to {\mathbb Z}/2 \to 0 \]

given by multiplication by 2.

Part of my “philosophy” for the book, if that is not too grand a word, is to present things on an “as needed” basis. So, right here we need the Pontrjagin classes and nothing else, so that’s what I sketch. later on we’ll have to come back and talk about Euler and Stiefel-Whitney as well. This means some backtracking, but I hope it makes the material more digestible.

]]>To further enrich this sectarian dispute, there are other categories of manifolds (other religions, one might almost say), among which the category of *piecewise linear* or PL manifolds is the most well-known. A PL manifold can be defined (analogously to the smooth case) as a topological manifold with an atlas whose transition functions are PL homeomorphisms, or equivalently as a polyhedron such that each point has a neighborhood PL homeomorphic to Euclidean space. True believers from both the smooth and topological worlds will mock the clunky, angular nature of these combinatorial objects. On the other hand, if you want actually to represent a manifold (or anything else) on a computer you will eventually end up with something combinatorial, so why not embrace it? Several views on these basic questions of PL theology can be found in this MathOverflow post.

In keeping with its amateurish nature, the book will concentrate on the smooth and topological cases (an amateur is someone coming from outside, and as far as I can see, by the time you have started seriously studying PL topology you are no longer an outsider). The PL case is “in between” smooth and topological: this is because of a theorem of Whitehead, which says that every smooth manifold admits an (essentially unique) PL manifold structure. A recent exposition of this and some other basic results of PL topology is found in the lecture notes of Lurie.

One final remark: there are no (high-dimensional) exotic spheres in the PL or TOP categories. As we will see, the smooth exotic spheres \(\Sigma^7\) constructed by Milnor (say in dimension 7) arise by gluing two 7-disks along a diffeomorphism of their boundary 6-spheres. If this diffeomorphism extended over one of the 7-disks, the resulting \(\Sigma^7\) would clearly be diffeomorphic to the standard \(S^7\). But in the topological or PL categories, it is trivial to make such an extension: we just “form the cone”, in other words, work in polar coordinates, map the radial coordinate to itself, and let the action on the boundary sphere determine the angular coordinate. It is a surprising fact, then, that the techniques of surgery theory, developed to study exotic spheres in the smooth category, provide nontrivial information also in the PL and topological categories, where no such examples exist.

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In this installment, we introduce the main objects that surgery theory is about – manifolds, especially (the reason for this will become apparent later) manifolds of *high dimension*, which typically means dimension at least five. Even as regards the definitions, though, there are important distinctions to be made. The most natural notion of a manifold is simply “a (nice) topological space locally homeomorphic to Euclidean space” – that is, a *topological* manifold. Historically, however, manifolds arose in connection with problems in differential geometry and analysis and thus possessed not just a topological but a *smooth* structure (i.e., it makes sense to talk about infinitely differentiable functions on such a manifold, not just about continuous ones.) One might naturally suppose that a topological manifold can always be “smoothed out”, just as a continuous function can always be approximated by a differentiable one. However, this is not the case. In fact, our story really gets its start in the middle fifties, when Milnor discovers that \( S^7 \) admits several *distinct* differentiable structures.

I am very interested to hear how accessible readers find this chapter. Could you follow the presentation/do the exercises? I should mention that I learned some of the ideas in Chapter 1 from some lecture notes of Tom Farrell, “Introduction to High-Dimensional Manifold Topology”, 2001.

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