# An introduction to the category of spectra

N. P. Strickland

These notes give a brief introduction to the category of spectra as defined in
stable homotopy theory. In particular, Section 5 discusses an extensive list
of examples of spectra whose properties have been found to be interesting.
Although many references are given, most of them are old. A survey of more
recent research and exposition would be valuable, but is not attempted here.

##  1\. Introduction

Early in the history of homotopy theory, people noticed a number of phenomena
suggesting that it would be convenient to work in a context where one could
make sense of negative-dimensional spheres. Let $X$ be a finite pointed
simplicial complex; some of the relevant phenomena are as follows.

  * •

For most $n$, the homotopy sets $\pi_{n}X$ are abelian groups. The proof
involves consideration of $S^{n-2}$ and so breaks down for $n<2$; this would
be corrected if we had negative spheres.

  * •

Calculation of homology groups is made much easier by the existence of the
suspension isomorphism $\widetilde{H}_{n+k}\Sigma^{k}X=\widetilde{H}_{n}X$.
This does not generally work for homotopy groups. However, a theorem of
Freudenthal says that if $X$ is a finite complex, we at least have a
suspension isomorphism $\pi_{n+k}\Sigma^{k}X=\pi_{n+k+1}\Sigma^{k+1}X$ for
large $k$. If could work in a context where $S^{-k}$ makes sense, we could
smash everything with $S^{-k}$ to get a suspension isomorphism in homotopy
parallel to the one in homology.

  * •

We can embed $X$ in $S^{k+1}$ for large $k$, and let $Y$ be the complement.
Alexander duality says that $\widetilde{H}_{n}Y=\widetilde{H}^{k-n}X$, showing
that $X$ can be “turned upside-down”, in a suitable sense. The shift by $k$ is
unpleasant, because the choice of $k$ is not canonical, and the minimum
possible $k$ depends on $X$. Moreover, it is unsatisfactory that the homotopy
type of $Y$ is not determined by that of $X$ (even after taking account of
$k$). In a context where negative spheres exist, one can define
$DX=S^{-k}\wedge Y$; one finds that $\widetilde{H}_{n}DX=\widetilde{H}^{-n}X$
and that $DX$ is a well-defined functor of $X$, in a suitable sense.

  * •

The Bott periodicity theorem says that the homotopy groups of the infinite
orthogonal group $O(\infty)$ satisfy $\pi_{k+8}O(\infty)=\pi_{k}O(\infty)$ for
all $k\geq 0$. It would be pleasant and natural to extend this pattern to
negative values of $k$, which would again require negative spheres.

Considerations such as these led to the construction of the Spanier-Whitehead
category ${\mathcal{F}}$ of finite spectra, which we briefly survey in Section
2. Although fairly straightforward, and very beautiful and interesting, this
category has two defects.

  * •

Many of the most important examples in homotopy theory are infinite complexes:
Eilenberg-MacLane spaces, classifying spaces of finite groups, infinite-
dimensional grassmannians and so on. The category ${\mathcal{F}}$ is strongly
tied to finite complexes, so a wider framework is needed to capture these
examples.

  * •

Ordinary homotopy theory is made both easier and more interesting by its
connections with geometry. However, ${\mathcal{F}}$ is essentially a
homotopical category, with no geometric structure behind it. This also
prevents a good theory of spectra with a group action, or of bundles of
spectra over a space, or of diagrams of spectra.

The first problem was addressed by a number of people, but the definitive
answer was provided by Boardman. He constructed a category ${\mathcal{B}}$
with excellent formal properties parallel to those of ${\mathcal{F}}$, whose
subcategory of finite objects (suitably defined) is equivalent to
${\mathcal{F}}$. A popular exposition of this category is in Adams’ book
[ad:shg]. Margolis [ma:ssa] gave a list of the main formal properties of
${\mathcal{B}}$ and its relationship with ${\mathcal{F}}$. He conjectured
(with good evidence) that they characterise ${\mathcal{B}}$ up to equivalence.
See [chst:pmh] for some new evidence for this conjecture, and [hopast:ash] for
an investigation of some related systems of axioms.

The second problem took much longer to resolve. There have been a number of
constructions of topological categories whose associated homotopy category
(suitably defined) is equivalent to ${\mathcal{B}}$, with steadily improving
formal properties [lemast:esh, el:ggs, ekmm:rma, hoshsm:ss]. There is also a
theorem of Lewis [le:ccs] which shows that it is impossible to have all the
good properties that one might naively hope for. We will sketch one
construction in Section 4.

##  2\. The finite stable category

###  2.1. Basics

We first recall some basic definitions. In this section all spaces are assumed
to be finite CW complexes with basepoints. (We could equally well use
simplicial complexes instead, at the price of having to subdivide and
simplicially approximate from time to time.) We write $0$ for all basepoints,
and we write $[A,B]$ for the set of based homotopy classes of maps from $A$ to
$B$. We define $A\vee B$ to be the quotient of the disjoint union of $A$ and
$B$ in which the two basepoints are identified together. We also define
$A\wedge B$ to be the quotient of $A\times B$ in which the subspace $A\times
0\cup 0\times B$ is identified with the single point $(0,0)$. This is called
the smash product of $A$ and $B$; note that it is commutative and associative
up to isomorphism and that $S^{0}\wedge A=A$, where $S^{0}=\\{0,1\\}$.
Moreover, $A\wedge(B\vee C)=(A\wedge B)\vee(A\wedge C)$ and $[A\vee
B,C]=[A,C]\times[B,C]$ (so $\vee$ is the coproduct in the homotopy category of
pointed spaces). We let $S^{1}$ denote the quotient of $[0,1]$ in which $0$ is
identified with $1$, and we write $\Sigma A=S^{1}\wedge A$. One can check that
$\Sigma$ sends the ball $B^{n}$ to $B^{n+1}$ and the sphere $S^{n}$ to
$S^{n+1}$, and the reduced homology of $\Sigma A$ is just
$\widetilde{H}_{n}(\Sigma A)=\widetilde{H}_{n-1}(A)$. Thus, we think of
$\Sigma$ as shifting all dimensions by one.

The quotient space of $S^{1}$ in which $1/2$ is identified with $0$ is
evidently homeomorphic to $S^{1}\vee S^{1}$, so we get a map $\delta\colon
S^{1}\xrightarrow{}S^{1}\vee S^{1}$ and thus a map $\delta\colon\Sigma
A\xrightarrow{}\Sigma A\vee\Sigma A$. It is well-known that the induced map
$\delta^{*}\colon[\Sigma A,B]\times[\Sigma A,B]\xrightarrow{}[\Sigma A,B]$
makes $[\Sigma A,B]$ into a group. There are apparently $n$ different group
structures on $[\Sigma^{n}A,B]$, but it is also well-known that they are all
the same, and they are commutative when $n>1$. We have an evident sequence of
maps

$[A,B]\xrightarrow{\Sigma}[\Sigma A,\Sigma
B]\xrightarrow{\Sigma}[\Sigma^{2}A,\Sigma^{2}B]\xrightarrow{}\ldots.$

Apart from the first two terms, it is a sequence of Abelian groups and
homomorphisms. By a fundamental theorem of Freudenthal, after a finite number
of terms, it becomes a sequence of isomorphisms. We define
$[\Sigma^{\infty}A,\Sigma^{\infty}B]$ to be the group
$[\Sigma^{N}A,\Sigma^{N}B]$ for large $N$, or if you prefer the colimit
$\operatornamewithlimits{\underset{\longrightarrow}{lim}}_{N}[\Sigma^{N}A,\Sigma^{N}B]$.
After doing a little point-set topology, one concludes that this is the same
as the set $[A,QB]$, where
$QB=\operatornamewithlimits{\underset{\longrightarrow}{lim}}_{N}\Omega^{N}\Sigma^{N}B$
and $\Omega^{N}C$ means the space of based continuous maps
$S^{N}\xrightarrow{}C$, with a suitable topology.

###  2.2. Finite spectra

One can define a category with one object called $\Sigma^{\infty}A$ for each
finite CW complex $A$, and morphisms $[\Sigma^{\infty}A,\Sigma^{\infty}B]$. It
is easy to see that $\Sigma$ induces a full and faithful endofunctor of this
category. We prefer to arrange things so that $\Sigma$ is actually an
equivalence of categories. Accordingly, we define a category ${\mathcal{F}}$
whose objects are expressions of the form $\Sigma^{\infty+n}A$ where $A$ is a
finite CW complex and $n$ is an integer. (If you prefer, you can take the
objects to be pairs $(n,A)$.) We refer to these objects as finite spectra. The
maps are

$[\Sigma^{\infty+n}A,\Sigma^{\infty+m}B]=\operatornamewithlimits{\underset{\longrightarrow}{lim}}_{N}[\Sigma^{N+n}A,\Sigma^{N+m}B].$

Freudenthal’s theorem again assures us that the limit is attained at a finite
stage. The functor $\Sigma$ induces a self-equivalence of the category
${\mathcal{F}}$. There are evident extensions of the functors $\vee$ and
$\wedge$ to ${\mathcal{F}}$ such that
$\Sigma^{\infty}A\vee\Sigma^{\infty}B=\Sigma^{\infty}(A\vee B)$ and
$\Sigma^{\infty}A\wedge\Sigma^{\infty}B=\Sigma^{\infty}(A\wedge B)$ (although
care is needed with signs when defining the smash product of morphisms). The
category ${\mathcal{F}}$ is additive, with biproduct given by the functor
$\vee$. The morphism sets $[X,Y]$ in ${\mathcal{F}}$ are finitely generated
Abelian groups. One can define homology of finite spectra by
$H_{n}\Sigma^{\infty+m}A=\widetilde{H}_{n-m}A$, and then the map

$H_{*}\colon{\mathbb{Q}}\otimes[X,Y]\xrightarrow{}\prod_{n}\operatorname{Hom}(H_{n}(X;{\mathbb{Q}}),H_{n}(Y;{\mathbb{Q}}))$

is an isomorphism. The groups $[X,Y]$ themselves are known to be recursively
computable, but the guaranteed algorithms are of totally infeasible
complexity. Nonetheless, there are methods of computation which require more
intelligence than the algorithms but have a reasonable chance of success.

###  2.3. Stable homotopy groups of spheres

Even the groups $\pi_{n}^{S}=[\Sigma^{\infty+n}S^{0},\Sigma^{\infty}S^{0}]$
are hard, and are only known for $n\leq 60$ or so (they are zero when $n<0$).
The first few groups are as follows:

$\displaystyle\pi_{0}^{S}$ $\displaystyle={\mathbb{Z}}\\{\iota\\}$
$\displaystyle\pi_{1}^{S}$ $\displaystyle=\mathbb{Z}/2\\{\eta\\}$
$\displaystyle\pi_{2}^{S}$ $\displaystyle=\mathbb{Z}/2\\{\eta^{2}\\}$
$\displaystyle\pi_{3}^{S}$ $\displaystyle=\mathbb{Z}/24\\{\nu\\}$
$\displaystyle\pi_{4}^{S}$ $\displaystyle=0$ $\displaystyle\pi_{5}^{S}$
$\displaystyle=0$ $\displaystyle\pi_{6}^{S}$
$\displaystyle=\mathbb{Z}/2\\{\nu^{2}\\}$ $\displaystyle\pi_{7}^{S}$
$\displaystyle=\mathbb{Z}/240\\{\sigma\\}$

Here $\iota$ is the identity map, and $\eta$ comes from the map

$\eta\colon
S^{3}=\\{(z,w)\in{\mathbb{C}}^{2}\;|\;|z|^{2}+|w|^{2}=1\\}\xrightarrow{}{\mathbb{C}}\cup\\{\infty\\}=S^{2}$

defined by $\eta(z,w)=z/w$. Similarly, $\nu$ comes from division of
quaternions, and $\sigma$ from division of octonions (but most later groups
cannot be described in a similarly explicit way). The expression $\eta^{2}$
really means $\eta\circ(\Sigma\eta)$, and $\nu^{2}$ means
$\nu\circ(\Sigma^{3}\nu)$.

Many general results are also known. For example, for any prime $p$, the
$p$-torsion part of $\pi_{n}^{S}$ is known for $n<2p^{3}-2p$ and is zero for
$n<2p-3$ (provided $n\neq 0$). Both the rank and the exponent are finite but
unbounded as $n$ tends to infinity. The group $\pi_{*}^{S}$ is a graded ring,
and is commutative in the graded sense. An important theorem of Nishida says
that all elements of degree greater than zero are nilpotent.

###  2.4. Triangulation

The category ${\mathcal{F}}$ is not Abelian. Instead, it has a triangulated
structure. This means that there is a distinguished class of diagrams of the
shape $X\xrightarrow{f}Y\xrightarrow{g}Z\xrightarrow{h}\Sigma X$ (called exact
triangles) with certain properties to be listed below. In our case the exact
triangles can be described as follows. Let $A$ be a subcomplex of a finite CW
complex $B$, and let $C$ be obtained from $B$ by attaching a cone $I\wedge A$
along the subspace $\\{1\\}\times A=A$. There is an evident copy of $B$ in
$C$, and if we collapse it to a point we get a copy of $\Sigma A$. We thus
have a diagram of spaces $A\xrightarrow{}B\xrightarrow{}C\xrightarrow{}\Sigma
A$. We say that a diagram $X\xrightarrow{}Y\xrightarrow{}Z\xrightarrow{}\Sigma
X$ of finite spectra is an exact triangle if it is isomorphic to a diagram of
the form
$\Sigma^{\infty+n}A\xrightarrow{}\Sigma^{\infty+n}B\xrightarrow{}\Sigma^{\infty+n}C\xrightarrow{}\Sigma^{\infty+n+1}A$
for some $n\in{\mathbb{Z}}$ and some $A$, $B$ and $C$ as above. Incidentally,
one can show that $C$ is homotopy equivalent to the space $B/A$ obtained from
$B$ by identifying $A$ with the basepoint.

The axioms for a triangulated category are as follows. In our case, they all
follow from the theory of Puppe sequences in unstable homotopy theory.

  * (a)

Any diagram isomorphic to an exact triangle is an exact triangle.

  * (b)

Any diagram of the form $0\xrightarrow{}X\xrightarrow{1}X\xrightarrow{}\Sigma
0=0$ is an exact triangle.

  * (c)

Any diagram $X\xrightarrow{f}Y\xrightarrow{g}Z\xrightarrow{h}\Sigma X$ is an
exact triangle if and only if the diagram
$Y\xrightarrow{g}Z\xrightarrow{h}\Sigma X\xrightarrow{-\Sigma f}\Sigma X$ is
an exact triangle.

  * (d)

For any map $f\colon X\xrightarrow{}Y$ there exists a spectrum $Z$ and maps
$g,h$ such that $X\xrightarrow{f}Y\xrightarrow{g}Y\xrightarrow{h}\Sigma X$ is
an exact triangle.

  * (e)

Suppose we have a diagram as shown below (with $h$ missing), in which the rows
are exact triangles and the rectangles commute. Then there exists a
(nonunique) map $h$ making the whole diagram commutative.

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\Sigma
U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Sigma
f}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Sigma
X}$

  * (f)

Suppose we have maps $X\xrightarrow{v}Y\xrightarrow{u}Z$, and exact triangles
$(X,Y,U)$, $(X,Z,V)$ and $(Y,Z,W)$ as shown in the diagram. (A circled arrow
$U\longrightarrow\circ\hskip 8.00003ptX$ means a map $U\xrightarrow{}\Sigma
X$.) Then there exist maps $r$ and $s$ as shown, making $(U,V,W)$ into an
exact triangle, such that the following commutativities hold:

$au=rd\qquad\qquad es=(\Sigma v)b\qquad\qquad sa=f\qquad\qquad br=c$
$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\scriptstyle{b}$$\scriptstyle{\bigcirc}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{uv}$$\scriptstyle{v}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{f}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r}$$\scriptstyle{c}$$\scriptstyle{\bigcirc}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{u}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{e}$$\scriptstyle{\bigcirc}$

The last axiom is called the octahedral axiom (the diagram can be turned into
an octahedron by lifting the outer vertices and drawing an extra line from $W$
to $U$). In our case it basically just says that when we have inclusions
$A\subseteq B\subseteq C$ of CW complexes we have $(C/A)/(B/A)=C/B$.

One of the most important consequences of the axioms is that whenever
$X\xrightarrow{}Y\xrightarrow{}Z\xrightarrow{}\Sigma X$ is an exact triangle
and $W$ is a finite spectrum, we have long exact sequences

$\ldots\xrightarrow{}[W,\Sigma^{-1}Z]\xrightarrow{}[W,X]\xrightarrow{}[W,Y]\xrightarrow{}[W,Z]\xrightarrow{}[W,\Sigma
X]\xrightarrow{}\ldots$

and

$\ldots\xleftarrow{}[\Sigma^{-1}Z,W]\xleftarrow{}[X,W]\xleftarrow{}[Y,W]\xleftarrow{}[Z,W]\xleftarrow{}[\Sigma
X,W]\xleftarrow{}\ldots.$

###  2.5. Thom spectra

Let $X$ be a finite CW complex, and let $V$ be a vector bundle over $X$. The
Thom space $X^{V}$ can be defined as the one-point compactification of the
total space of $V$. This has many interesting properties, not least of which
is the fact that when $V$ is an oriented bundle of dimension $n$, the reduced
cohomology $\widetilde{H}^{*}(X^{V})$ is a free module over $H^{*}(X)$ on one
generator in dimension $n$. This construction can be generalised to virtual
bundles, in other words formal expressions of the form $V-W$, except that we
now have a Thom spectrum $X^{V-W}$ rather than a Thom space. The construction
is to choose a surjective map from a trivial bundle ${\mathbb{R}}^{n}\times X$
onto $W$, with kernel $U$ say, and define $X^{V-W}=\Sigma^{\infty-n}X^{V\oplus
U}$.

###  2.6. Duality

For any finite spectrum $X$, there is an essentially unique spectrum $DX$
(called the Spanier-Whitehead dual of $X$) equipped with a natural isomorphism
$[W\wedge X,Y]=[W,DX\wedge Y]$. This can be constructed in a number of
different ways. One way is to start with a simplicial complex $A$ and embed it
simplicially as a proper subcomplex of $S^{N+1}$ for some $N>0$. One can show
that the complement of $A$ has a deformation retract $B$ which is a finite
simplicial complex, and $D\Sigma^{\infty+n}A=\Sigma^{\infty-n-N}B$. Note that
Alexander duality implies that $H_{m}X=H^{-m}DX$.

An important example arises when $X=\Sigma^{\infty}M_{+}$ for some smooth
manifold $M$, with tangent bundle $\tau$ say. It is not hard to show
geometrically that $D(\Sigma^{\infty}M_{+})$ is $M^{-\tau}$, the Thom spectrum
of the virtual bundle $-\tau$ over $M$; this phenomenon is called _Atiyah
duality_.

We also write $F(X,Y)=DX\wedge Y$. This is a functor in both variables, it
preserves cofibrations up to sign, and the defining property of $DX$ can be
rewritten as $[W,F(X,Y)]=[W\wedge X,Y]$.

###  2.7. Splittings

It often happens that we have a finite complex $X$ that cannot be split into
simpler pieces, but that the finite spectrum $\Sigma^{\infty}X$ does have a
splitting. Group actions are one fruitful source of splittings. If a finite
group $G$ acts on $X$, then the map
$G\xrightarrow{}\operatorname{Aut}(\Sigma^{\infty}X)$ extends to a ring map
${\mathbb{Z}}[G]\xrightarrow{}\operatorname{End}(\Sigma^{\infty}X)$. If
$\widetilde{H}_{*}X$ is a $p$-torsion group, then this will factor through
$({\mathbb{Z}}/p^{n})[G]$ for large $n$. Any idempotent element in
$({\mathbb{Z}}/p)[G]$ can be lifted uniquely to an idempotent in
$({\mathbb{Z}}/p^{n})[G]$, which will give an idempotent in
$\operatorname{End}(\Sigma^{\infty}X)$ and thus a splitting of $X$. The
methods of modular representation theory give good information about
idempotents in group rings, and thus a supply of interesting splittings. The
Steinberg idempotent in $({\mathbb{Z}}/p)[GL_{n}({\mathbb{Z}}/p)]$ gives
particularly important examples [mi:osm, mipr:ssd], as do various idempotents
in $({\mathbb{Z}}/p)[\Sigma_{n}]$ (see [ra:nps]*Appendix C).

Another common situation is to have a finite complex $X$ and a filtration
$F_{0}X\subseteq F_{1}X\subseteq\ldots\subseteq F_{n}X=X$ that splits stably,
giving an equivalence
$\Sigma^{\infty}X\simeq\bigvee_{n}\Sigma^{\infty}F_{n}X/F_{n-1}X$ of finite
spectra. For example, one can take $X=U(m)$, and let $F_{n}X$ be the space of
matrices $A\in U(m)$ for which the rank of $A-I$ is at most $n$. A theorem of
Miller [mi:sss] says that the filtration splits stably, and that the quotient
$F_{n}U(m)/F_{n-1}U(m)$ is the Thom space of a certain bundle over the
Grassmannian of $n$-planes in ${\mathbb{C}}^{m}$. Later we will explain how to
interpret $\Sigma^{\infty}X$ when $X$ is an infinite complex; there are many
examples in which $X$ has a stably split filtration in which the quotients are
finite spectra. This holds for $X=BU(n)$ or $X=\Omega U(n)$ or
$X=\Omega^{n}S^{n+m}$, for example. The splitting of $\Omega^{n}S^{n+m}$ is
due to Snaith [sn:sdo]; the Snaith summands in $\Omega^{2}S^{3}$ are called
_Brown-Gitler spectra_ , and they have interesting homological properties with
many applications [brgi:scc, sc:ums, huku:csu].

##  3\. Cobordism and Morava $K$-theory

We next outline the theory of complex cobordism [ra:ccs], and the results of
Hopkins, Devinatz and Smith showing how complex cobordism reveals an important
part of the structure of ${\mathcal{F}}$.

Given a space $X$ and an integer $n\geq 0$ we define a _geometric $n$-chain_
in $X$ to be a compact smooth $n$-manifold $M$ (possibly with boundary)
equipped with a continuous map $f\colon M\xrightarrow{}X$. We regard
$(M_{0},f_{0})$ and $(M_{1},f_{1})$ as equivalent if there is a
differomorphism $g\colon M_{0}\xrightarrow{}M_{1}$ with $f_{1}g=f_{0}$. We
write $GC_{n}X$ for the set of equivalence classes, which is a commutative
semigroup under disjoint union. We define a differential $\partial\colon
GC_{n}X\xrightarrow{}GC_{n-1}X$ by $\partial[M,f]=[\partial M,f|_{\partial
M}]$. One can make sense of the homology $MO_{*}X=H(GC_{*}X,\partial)$, and
(because $\partial(M\times I)=M\amalg M$) one finds that $MO_{*}X$ is a vector
space over ${\mathbb{Z}}/2$. In the case where $X$ is a point, one can use
cartesian products to make $MO_{*}(\operatorname{point})$ into a graded ring,
which is completely described by a remarkable theorem of Thom: it is a
polynomial algebra over ${\mathbb{Z}}/2$ with one generator $x_{n}$ in degree
$n$ for each integer $n>0$ not of the form $2^{k}-1$. New perspectives on this
answer and the underlying algebra were provided by Quillen [qu:fgl, qu:epc]
and Mitchell [mi:psm]. One can also show that
$MO_{*}X=MO_{*}(\operatorname{point})\otimes_{{\mathbb{Z}}/2}H_{*}(X;{\mathbb{Z}}/2)$,
so this construction does not yield new invariants of spaces. This isomorphism
gives an obvious way to define $MO_{*}X$ when $X$ is a finite spectrum.

The story changes however, if we work with oriented manifolds. This gives
groups $MSO_{*}X$ with a richer structure; in particular, they are not
annihilated by $2$. There are various hints that complex manifolds would give
still more interesting invariants, but there are technical problems, not least
the lack of a good theory of complex manifolds with boundary. It turns out to
be appropriate to generalize and consider manifolds with a specified complex
structure on the stable normal bundle, known as “stably complex manifolds”.
The precise definitions are delicate; details are explained in [qu:epc], and
Buchstaber and Ray [bura:tst] have provided naturally occuring examples where
the details are important. In any case, one ends up with a ring
$MU_{*}=MU_{*}(\operatorname{point})$, and groups $MU_{*}X$ for all spaces $X$
that are modules over it. It is not the case that
$MU_{*}X=MU_{*}\otimes_{\mathbb{Z}}H_{*}X$, but there is still a suspension
isomorphism, which allows one to define $MU_{*}X$ when $X$ is a finite
spectrum. One finds that this is a generalized homology theory (known as
_complex cobordism_), so it converts cofibre seqences of spectra to long exact
sequences of modules. The nilpotence theory of Devinatz, Hopkins and Smith
[dehosm:nsh, hosm:nshii, ra:nps] (which will be outlined below) shows that
$MU_{*}X$ is an extremely powerful invariant of $X$.

The ring $MU_{*}$ turns out to be a polynomial algebra over ${\mathbb{Z}}$
with one generator in each positive even degree. There is no canonical system
of generators, but nonetheless, Quillen showed that $MU_{*}$ is canonically
isomorphic to an algebraically defined object: Lazard’s classifying ring for
formal group laws [ad:shg, ra:ccs]. This was the start of an extensive
relationship between stable homotopy and formal group theory. The algebra
provides many natural examples of graded rings $A_{*}$ equipped with a formal
group law and thus a map $MU_{*}\xrightarrow{}A_{*}$. It is natural to ask
whether there is a generalised homology theory $A_{*}(X)$ whose value on a
point is the ring $A_{*}$. Satisfactory answers for a broad class of rings
$A_{*}$ are given in [st:pmm], which surveys and consolidates a great deal of
much older literature and extends newer ideas from [ekmm:rma]. In particular,
we can consider the rings $K(p,n)_{*}={\mathbb{F}_{p}}[v_{n},v_{n}^{-1}]$,
where $p$ is prime and $n>0$ and $v_{n}$ has degree $2(p^{n}-1)$. This is made
into an algebra over $MU_{*}$ using a well-known formal group law, and by old
or new methods, one can construct an associated generalised homology theory
$K(p,n)_{*}X$, known as _Morava $K$-theory_. It is convenient to extend the
definition by putting $K(p,0)_{*}X=H_{*}(X;{\mathbb{Q}})$.

A key theorem of Hopkins, Devinatz and Smith says that if
$f\colon\Sigma^{d}X\xrightarrow{}X$ is a self-map of a finite spectrum $X$,
and $K(p,n)_{*}(f)=0$ for all primes $p$ and all $n\geq 0$, then the iterated
composite $f^{m}\colon\Sigma^{md}X\xrightarrow{}X$ is zero for large $m$, or
in other words $f$ is composition-nilpotent. They also show that Morava
$K$-theory detects nilpotence in a number of other senses, and give
formulations involving the single theory $MU$ instead of the collection of
theories $K(p,n)$.

If $R$ is a commutative ring, it is well-known that the ideal of nilpotent
elements is the intersection of all the prime ideals, so the Zariski spectrum
is unchanged if we take the quotient by this ideal. One can deduce that the
classification of certain types of subcategories of the abelian category of
$R$-modules is again insensitive to the ideal of nilpotents. Our category
${\mathcal{F}}$ of finite spectra is triangulated rather than abelian, but
nonetheless Hopkins and Smith developed an analogous theory and deduced a
classification of the thick subcategories of ${\mathcal{F}}$, with many
important consequences.

##  4\. Boardman’s category ${\mathcal{B}}$

It is clearly desirable to have a category ${\mathcal{C}}$ analogous to
${\mathcal{F}}$ but without finiteness conditions. There are various obvious
candidates: one could take the Ind-completion of ${\mathcal{F}}$, or just
follow the definition of ${\mathcal{F}}$ but allow infinite CW complexes
instead of finite ones. Unfortunately, these categories turn out to have
unsatisfactory technical properties. The requirements were first assembled in
axiomatic form by Margolis [ma:ssa]; in outline, they are as follows:

  * •

${\mathcal{C}}$ should be a triangulated category

  * •

Every family $\\{X_{\alpha}\\}$ of objects in ${\mathcal{C}}$ should have a
coproduct, written $\bigvee_{\alpha}X_{\alpha}$

  * •

For any $X,Y\in{\mathcal{C}}$ there should be functorially associated objects
$X\wedge Y$ and $F(X,Y)$ making ${\mathcal{C}}$ a closed symmetric monoidal
category.

  * •

If we let $\text{small}({\mathcal{C}})$ be the subcategory of objects $W$ for
which the natural map
$\bigoplus_{\alpha}[W,X_{\alpha}]\xrightarrow{}[W,\bigvee_{\alpha}X_{\alpha}]$
is always an isomorphism, then $\text{small}({\mathcal{C}})$ should be
equivalent to ${\mathcal{F}}$.

Historically, the work of Margolis came after Boardman’s construction of a
category ${\mathcal{B}}$ satisfying the axioms, and Adams’s explanation
[ad:shg] of a slightly different way to approach the construction. Margolis
conjectured that if ${\mathcal{C}}$ satisfies the axioms then ${\mathcal{C}}$
is equivalent to ${\mathcal{B}}$. Schwede and Shipley [scsh:uts] have proved
that this is true, provided that ${\mathcal{C}}$ is the homotopy category of a
closed model category in the sense of Quillen [qu:ha, dwsp:htm] satisfying
suitable axioms. There is also good evidence for the conjecture without this
additional assumption. The objects of ${\mathcal{B}}$ are generally called
_spectra_ , although in some contexts one introduces different words to
distinguish between objects in different underlying geometric categories.

Probably the best approach to constructing ${\mathcal{B}}$ is via the theory
of _orthogonal spectra_ , as we now describe [mamash:mcd]. Let ${\mathcal{V}}$
denote the category of finite-dimensional vector spaces over ${\mathbb{R}}$
equipped with an inner product. The morphisms are linear isomorphisms that
preserve inner products. For any $V\in{\mathcal{V}}$ (with $\dim(V)=n$ say) we
write $S^{V}$ for the one-point compactification of $V$; this is homeomorphic
to $S^{n}$. An orthogonal spectrum $X$ consists of a functor
${\mathcal{V}}\to\\{\text{based spaces}\\}$ together with maps $S^{U}\wedge
X(V)\to X(U\oplus V)$ satisfying various continuity and compatibility
conditions that we will not spell out. We write ${\mathcal{S}}$ for the
category of orthogonal spectra. For example, a based space $A$ gives an
orthogonal spectrum $\Sigma^{\infty}A$ with $(\Sigma^{\infty}A)(U)=S^{U}\wedge
A$. A vector space $T\in{\mathcal{V}}$ gives positive and negative sphere
spectra $S^{\pm T}$: the value of the sphere spectrum $S^{T}$ at
$U\in{\mathcal{V}}$ is the sphere space $S^{U}\wedge S^{T}=S^{U\oplus T}$, and

$S^{-T}(U)=\\{(\alpha,u)\;|\;\alpha\in{\mathcal{V}}(T,U),\;u\in U,\;\langle
u,\alpha(T)\rangle=0\\}\cup\\{\infty\\}.$

For orthogonal spectra $X$ and $Y$, the morphism set ${\mathcal{S}}(X,Y)$ has
a natural topology, and we could define an associated homotopy category by the
rule $[X,Y]=\pi_{0}{\mathcal{S}}(X,Y)$. Unfortunately, the resulting category
is not the one that we want. Instead, we define the homotopy groups of $X$ by
the rule
$\pi_{k}(X)=\operatornamewithlimits{\underset{\longrightarrow}{lim}}_{N}\pi_{k+N}(X({\mathbb{R}}^{N}))$.
We then say that a map $f\colon X\to Y$ is a _weak equivalence_ if
$\pi_{*}(f)\colon\pi_{*}(X)\to\pi_{*}(Y)$ is an isomorphism. We now construct
a new category $\operatorname{Ho}({\mathcal{S}})$ by starting with
${\mathcal{S}}$ and adjoining formal inverses for all weak equivalences. It
can be shown that this is equivalent to ${\mathcal{B}}$ (or can be taken as
the definition of ${\mathcal{B}}$).

This process of adjoining formal inverses can be subtle. To manage the
subtleties, we need the theory of model categories in the sense of Quillen
[dwsp:htm, ho:mc]. In particular, this will show that
$\operatorname{Ho}({\mathcal{S}})(X,Y)=\pi_{0}{\mathcal{S}}(X,Y)$ for certain
classes of spectra $X$ and $Y$; this is enough to get started with
computations and prove that Margolis’s axioms are satisfied.

Given orthogonal spectra $X$, $Y$ and $Z$, a _pairing_ from $X$ and $Y$ to $Z$
consists of maps $\alpha_{U,V}\colon X(U)\wedge Y(V)\to Z(U\oplus V)$
satisfying some obvious compatibility conditions. One can show that there is
an orthogonal spectrum $X\wedge Y$ such that pairings from $X$ and $Y$ to $Z$
biject with morphisms from $X\wedge Y$ to $Z$. Basic examples are that
$S^{0}\wedge X=X$ and
$\Sigma^{\infty}A\wedge\Sigma^{\infty}B=\Sigma^{\infty}(A\wedge B)$ and
$S^{U}\wedge S^{V}=S^{U\oplus V}$ and $S^{-U}\wedge S^{-V}=S^{-U-V}$. There is
a natural map $S^{U}\wedge S^{-U}\to S^{0}$ that is a weak equivalence but not
an isomorphism.

This construction gives a symmetric monoidal structure on ${\mathcal{S}}$.
This in turn gives rise to a symmetric monoidal structure on
$\operatorname{Ho}({\mathcal{S}})$; however, there are some hidden subtleties
in this step, which again are best handled by the general theory of model
categories. One consequence is that the topology of the classifying space of
the symmetric group $\Sigma_{k}$ is mixed in to the structure of the $k$-fold
smash product $X^{(k)}=X\wedge\dotsb\wedge X$ and the quotient
$X^{(k)}/\Sigma_{k}$.

A _ring spectrum_ is an object $R\in\operatorname{Ho}({\mathcal{S}})$ equipped
with a unit map $\eta\colon S^{0}\to R$ and a multiplication map $\mu\colon
R\wedge R\to R$ such that the following diagrams in
$\operatorname{Ho}({\mathcal{S}})$ commute:

---
$\textstyle{R\wedge R\wedge
R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu\wedge
1}$$\scriptstyle{1\wedge\mu}$$\textstyle{R\wedge
R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta\wedge
1}$$\scriptstyle{1}$$\textstyle{R\wedge
R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1\wedge\eta}$$\scriptstyle{1}$$\textstyle{R\wedge
R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{R}$$\textstyle{R}$

Because we now have a good underlying geometric category ${\mathcal{S}}$, we
can formulate a more precise notion: a _strict ring spectrum_ is an object
$R\in{\mathcal{S}}$ equipped with morphisms
$S^{0}\xrightarrow{\eta}R\xleftarrow{\mu}R\wedge R$ such that the above
diagrams commute in ${\mathcal{S}}$ (not just in
$\operatorname{Ho}({\mathcal{S}})$).

The symmetric monoidal structure on ${\mathcal{S}}$ includes a natural map
$\tau_{XY}\colon X\wedge Y\to Y\wedge X$. We say that a strict ring spectrum
$R$ is _strictly commutative_ if $\mu\circ\tau_{RR}=\mu$. This is a
surprisingly stringent condition, with extensive computational consequences. A
key point is that we have an iterated multiplication map
$R^{(k)}/\Sigma_{k}\to R$, and this brings into play the structure of
$R^{*}(B\Sigma_{k})$.

##  5\. Examples of spectra

Some important functors that construct objects of ${\mathcal{B}}$ are as
follows:

  * (a)

For any based space $X$ there is a suspension spectrum
$\Sigma^{\infty}X\in{\mathcal{B}}$, whose homotopy groups are given by
$\pi_{n}\Sigma^{\infty}X=\pi^{S}_{n}X=\operatornamewithlimits{\underset{\longrightarrow}{lim}}_{k}\pi_{n+k}\Sigma^{k}X$.
The relevant orthogonal spectrum is just $(\Sigma^{\infty}X)(V)=S^{V}\wedge
X$.

We will mention one important example of infinite complexes $X$ and $Y$ for
which $[\Sigma^{\infty}X,\Sigma^{\infty}Y]$ is well-understood. Let $G$ be a
finite group, with classifying space $BG$. Let $AG^{+}$ be the set of
isomorphism classes of finite sets with a $G$-action. We can define addition
and multiplication on $AG^{+}$ by $[X]+[Y]=[X\amalg Y]$ and $[X][Y]=[X\times
Y]$. There are no additive inverses, but we can formally adjoin them to get a
ring called $AG$, the Burnside ring of $G$; this is not hard to work with
explicitly. There is a ring map $\epsilon\colon AG\xrightarrow{}{\mathbb{Z}}$
defined by $\epsilon([X]-[Y])=|X|-|Y|$, with kernel $I$ say. We then have a
completed ring
$\widehat{A}G=\operatornamewithlimits{\underset{\longleftarrow}{lim}}_{n}AG/I^{n}$.
The Segal conjecture (which was proved by Carlsson [ca:esh]) gives an
isomorphism $\widehat{A}G\simeq[\Sigma^{\infty}BG_{+},\Sigma^{\infty}S^{0}]$.
One can deduce a description of
$[\Sigma^{\infty}BG_{+},\Sigma^{\infty}BH_{+}]$ in similar terms for any
finite group $H$.

  * (b)

For any virtual vector bundle $V$ over any space $X$, there is a Thom spectrum
$X^{V}\in{\mathcal{B}}$. In particular, if $V$ is the tautological virtual
bundle (of virtual dimension zero) over the classifying space $BU$, then there
is an associated Thom spectrum, normally denoted by $MU$. This has the
property that the groups $MU_{*}X$ (as in Section 3) are given by
$\pi_{*}(MU\wedge\Sigma^{\infty}X_{+})$ (this is proved by a geometric
argument, and is essentially the first step in the calculation of $MU_{*}$).
One can construct $MU$ (and also $MO$ and $MSO$) as strictly commutative ring
spectra.

Bott periodicity gives an equivalence $BU\simeq\Omega SU$. A range of
different proofs can be found in [mi:mt]*Section 23 [prse:lg]*Sections 6.4 and
8.8 [hu:fb]*Chapter 11 [be:npb]. The filtration of $\Omega SU$ by the
subspaces $\Omega SU(k)$ gives a filtration of $MU$ by subspectra $X(m)$,
which are important in the proof of the Hopkins-Devinatz-Smith nilpotence
theorem. There are models of these homotopy types that are strict ring
spectra, but they cannot be made commutative.

  * (c)

For any generalized cohomology theory $A^{*}$, there is an essentially unique
spectrum $A\in{\mathcal{B}}$ with $A^{n}X=[\Sigma^{\infty}X_{+},\Sigma^{n}A]$
for all spaces $X$ and $n\in{\mathbb{Z}}$. Similarly, for any generalized
homology theory $B_{*}$, there is an essentially unique spectrum
$B\in{\mathcal{B}}$ with $B_{n}X=\pi_{n}(B\wedge\Sigma^{\infty}X_{+})$ for all
spaces $X$ and $n\in{\mathbb{Z}}$. These facts are known as _Brown
representability_ ; the word “essentially” hides some subtleties.

  * (d)

In particular, for any abelian group $A$ there is an Eilenberg-MacLane
spectrum $HA\in{\mathcal{B}}$ such that
$[\Sigma^{\infty}X_{+},\Sigma^{n}HA]=H^{n}(X;A)$ and
$\pi_{n}(HA\wedge\Sigma^{\infty}X_{+})=H_{n}(X;A)$. (In fact, Brown
representability is not needed for this: we have an explicit orthogonal
spectram $HA$ where $HA(V)$ is the tensor product of $A$ with the free abelian
group generated by $V$, equipped with a suitable topology.) If $A$ is a
commutative ring, then $HA$ is a strictly commutative ring spectrum. It is
common to consider the case $A={\mathbb{Z}}/2$. Here it can be shown that

$\pi_{*}((H{\mathbb{Z}}/2)\wedge(H{\mathbb{Z}}/2))={\mathbb{Z}}/2[\xi_{1},\xi_{2},\xi_{3},\dotsc],$

with $|\xi_{k}|=2^{k}-1$. This is known as the _dual Steenrod algebra_ , and
denoted by ${\mathcal{A}}_{*}$. The dual group
${\mathcal{A}}^{k}=\operatorname{Hom}({\mathcal{A}}_{k},{\mathbb{Z}}/2)$ can
be identified with $[H{\mathbb{Z}}/2,\Sigma^{k}H{\mathbb{Z}}/2]$, so these
groups again form a graded ring (under composition), called the _Steenrod
algebra_. This ring is noncommutative, but its structure can be described
quite explicitly. It is important, because the mod $2$ cohomology of any space
(or spectrum) has a natural structure as an ${\mathcal{A}}^{*}$-module. There
is a similar story for mod $p$ cohomology when $p$ is an odd prime, but the
details are a little more complicated.

  * (e)

Another consequence of Brown representability is that there is a spectrum
$I\in{\mathcal{B}}$ such that
$[X,I]\simeq\operatorname{Hom}(\pi_{0}X,{\mathbb{Q}}/{\mathbb{Z}})$ for all
$X\in{\mathcal{B}}$. This is called the _Brown-Comenetz dual_ [brco:pdg] of
$S^{0}$; it is geometrically mysterious, and a fertile source of
counterexamples.

  * (f)

If $M_{*}$ is a flat module over $MU_{*}$, then the functor $X\mapsto
M_{*}\otimes_{MU_{*}}MU_{*}X$ is a homology theory, so there is a representing
spectrum $M$ with $\pi_{*}(M\wedge X)=M_{*}\otimes_{MU_{*}}MU_{*}X$. The
Landweber exact functor theorem shows that flatness is not actually necessary:
a weaker condition called _Landweber exactness_ will suffice [la:hpc]. This
condition is formulated in terms of formal group theory, and is often easy to
check in practice.

Often $M_{*}$ is a ring, and the $MU_{*}$-module structure arises from a ring
map $MU_{*}\to M_{*}$, which corresponds (by Quillen’s description of
$MU_{*}$) to a formal group law over $M_{*}$.

Important examples include the Johnson-Wilson spectra $E(p,n)$, with
$E(p,n)_{*}={\mathbb{Z}}_{(p)}[v_{1},\dotsc,v_{n}][v_{n}^{-1}]$ (where
$|v_{k}|=2(p^{k}-1)$). This has a canonical formal group law, which we will
not describe here. The ring $K(p,n)_{*}={\mathbb{Z}}/p[v_{n},v_{n}^{-1}]$ is
naturally a quotient $E(p,n)_{*}/I_{n}$, where
$I_{n}=(p,v_{1},\dotsc,v_{n-1})$. It it is not Landweber exact, but a
corresponding spectrum $K(p,n)$ can be constructed by other means. It is also
useful to consider the completed spectra $\widehat{E}(p,n)$ with

$\pi_{*}(\widehat{E}(p,n))=(E(p,n)_{*})^{\wedge}_{I_{n}}={\mathbb{Z}}_{p}[\\![v_{1},\dotsc,v_{n-1}]\\!][v_{n}^{\pm
1}].$

This is again Landweber exact.

When $M_{*}$ is a ring, one might hope to find a model of this homotopy type
that is actually a strict ring spectrum, preferably strictly commutative.
Unfortunately, this does not work very well. Often there will be uncountably
many different ways to make $M$ into a strict ring spectrum, with no way to
pick out a preferred choice. Moreover, there will often not be any choice that
is strictly commutative. However, by a theorem of Hopkins and Miller, there is
an essentially unique strictly commutative model for $\widehat{E}(p,n)$. The
reason why this case is special involves quite deep aspects of the algebraic
theory of formal groups.

If $M_{*}$ is an algebra over $MU_{*}$ that does not satisfy the Landweber
criterion, one can still try to produce a corresponding spectrum $M$ by other
methods. As mentioned previously, the paper [st:pmm] contains results in this
direction.

  * (g)

For any small symmetric monoidal category category ${\mathcal{A}}$, there is a
$K$-theory spectrum $K({\mathcal{A}})\in{\mathcal{B}}$. Computationally, this
is very mysterious, apart from the fact that it is always connective (ie
$\pi_{n}K({\mathcal{A}})=0$ for $n<0$) and $\pi_{0}K({\mathcal{A}})$ is the
group completion of the monoid of connected components in ${\mathcal{A}}$.
Thomason has shown [th:smc] that for every connective spectrum $X$ there
exists ${\mathcal{A}}$ with $K({\mathcal{A}})\simeq X$.

    * –

If ${\mathcal{A}}$ is the category of finite sets and isomorphisms, then
$K({\mathcal{A}})=\Sigma^{\infty}S^{0}$.

    * –

Let $G$ be a finite group, and let ${\mathcal{A}}$ be the category of finite
$G$-sets and isomorphisms. Let ${\mathcal{A}}_{f}$ (resp ${\mathcal{A}}_{t}$)
be the subcategory of free (resp. transitive) $G$-sets. Then
$K({\mathcal{A}})=\Sigma^{\infty}B({\mathcal{A}}_{t})_{+}$, which can also be
described as the wedge over the conjugacy classes of subgroups $H\leq G$ of
the spectra $\Sigma^{\infty}BW_{G}H_{+}$, or as the fixed point spectrum of
the $G$-equivariant sphere spectrum in the sense of Lewis-May-Steinberger
[lemast:esh]. On the other hand, $K({\mathcal{A}}_{f})=\Sigma^{\infty}BG_{+}$.

    * –

Work of Kathryn Lesh [le:fsa] can be interpreted as exhibiting symmetric
monoidal categories ${\mathcal{M}}_{n}$ (of “finite multisets with
multiplicities at most $n$”) whose $K$-theory is the $n$’th symmetric power of
$\Sigma^{\infty}S^{0}$.

    * –

If ${\mathcal{A}}$ is the symmetric monoidal category with object set
${\mathbb{N}}$ and only identity morphisms, then
$K({\mathcal{A}})=H{\mathbb{Z}}$.

    * –

One can set up a category ${\mathcal{A}}$, whose objects are smooth compact
closed $1$-manifolds, and whose morphisms are cobordisms between them. With
the right choice of details, the $K$-theory spectrum $K({\mathcal{A}})$ is
then closely related to the classifying space of the stable mapping class
group, and an important theorem of Madsen and Weiss [mawe:sms] can be
interpreted as saying that $K({\mathcal{A}})$ is the Thom spectrum of the
negative of the tautological bundle over ${\mathbb{C}}P^{\infty}$, up to
adjustment of $\pi_{-2}$.

    * –

If $R$ is a commutative ring and ${\mathcal{A}}$ is the category of finitely
generated projective $R$-modules, then $K({\mathcal{A}})$ is the algebraic
$K$-theory spectrum usually denoted by $K(R)$. Even in the case
$R={\mathbb{Z}}$, this contains a great deal of arithmetic information. By
rather different methods one can construct spectra called $THH(R)$ and $TC(R)$
(topological Hochschild homology and topological cyclic homology) that
approximate $K(R)$; there is an extensive literature on these approximations.
The definitions can be set up in such a way that the spectra $K(R)$, $THH(R)$
and $TC(R)$ are all strictly commutative ring spectra.

  * (h)

The above construction can be modified slightly to take account of a topology
on the morphism sets of ${\mathcal{A}}$. We can then feed in the category of
finite-dimensional complex vector spaces and isomorphisms (or a skeleton
thereof) to get a spectrum known as $kU$, the connective complex $K$-theory
spectrum, with a homotopy element $u\in\pi_{2}kU$ such that
$\pi_{*}kU={\mathbb{Z}}[u]$. This has the property that for finite complexes
$X$, the group $kU^{0}X=[\Sigma^{\infty}X_{+},kU]$ is the group completion of
the monoid of isomorphism classes of complex vector bundles on $X$, or in
other words the $K$-theory of $X$ as defined by Atiyah and Hirzebruch [at:kt]
(inspired by Grothendieck’s similar definition in the context of algebraic
geometry). By a colimit construction, one can build a periodized version
called $KU$ with $\pi_{*}KU={\mathbb{Z}}[u,u^{-1}]$. There are direct
constructions of $kU$ and $KU$ using Bott periodicity rather than symmetric
monoidal categories. There are also constructions with a more analytic
flavour, based on spaces of Fredholm operators and so on [jo:srs]. With the
right construction, both $kU$ and $KU$ are strictly commutative ring spectra.
It can be shown that $KU$ is naturally a Landweber exact $MU$-algebra, so
$KU_{*}(X)=KU_{*}\otimes_{MU_{*}}MU_{*}(X)$ for all $X$.

The infinite complex projective space ${\mathbb{C}}P^{\infty}$ is well-known
to be a commutative group up to homotopy. Using this, one can make the
spectrum $R=\Sigma^{\infty}({\mathbb{C}}P^{\infty})_{+}$ into a ring spectrum.
The standard identification ${\mathbb{C}}P^{1}=S^{2}$ gives rise to an element
$v\in\pi_{2}(R)$, and we can use a homotopy colimit construction to invert
this element, giving a new ring spectrum $R[v^{-1}]$. It is a theorem of
Snaith [sn:lsh] that $R[v^{-1}]$ is homotopy equivalent to $KU$.

  * (i)

Let $F$ be a functor from based spaces to based spaces. Under mild conditions,
we can use the homeomorphism $S^{1}\wedge S^{n}\xrightarrow{}S^{n+1}$ to get a
map

$S^{1}\xrightarrow{}\operatorname{Map}(S^{n},S^{n+1})\xrightarrow{F}\operatorname{Map}(FS^{n},FS^{n+1}),$

and thus an adjoint map $\Sigma FS^{n}\xrightarrow{}FS^{n+1}$. This gives a
sequence of spectra $\Sigma^{-n}\Sigma^{\infty}FS^{n}$, whose homotopy colimit
(in a suitable sense) is denoted by $D_{1}F$. This is called the
_linearization_ or _first Goodwillie derivative_ of $F$. Goodwillie [go:ci,
go:cii, go:ciii] has set up a “calculus of functors” in which the higher
derivatives are spectra $D_{n}F$ with an action of $\Sigma_{n}$. (The slogan
is that where the ordinary calculus of functions has a denominator of $n!$,
the calculus of functors will take coinvariants under an action of
$\Sigma_{n}$.) Even the derivatives of the identity functor are interesting;
they fit in an intricate web of relationships with partition complexes,
symmetric powers of the sphere spectrum, Steinberg modules, free Lie algebras
and so on. There are other versions of calculus for functors from other
categories to spaces, with applications to embeddings of manifolds, for
example.

  * (j)

A _Moore spectrum_ is a spectrum $X$ for which $\pi_{n}(X)=0$ when $n<0$ and
$H_{n}(X)=0$ when $n>0$. Let ${\mathcal{H}}$ be the category of Moore spectra.
The functor $H_{0}\colon{\mathcal{H}}\to\operatorname{Ab}$ is then close to
being an equivalence: for any $X,Y\in{\mathcal{H}}$ there is a natural short
exact sequence

$\operatorname{Ext}(H_{0}(X),H_{0}(Y))/2\xrightarrow{}[X,Y]\xrightarrow{}\operatorname{Hom}(\pi_{0}(X),\pi_{0}(Y)).$

Moreover, given any abelian group $A$ there is a Moore spectrum $SA$ (unique
up to non-canoncal isomorphism) with $H_{0}(SA)\simeq A$.

  * (k)

Let $C$ be an elliptic curve over a ring $k$. (Number theorists are often
interested in the case where $k$ is a small ring like ${\mathbb{Z}}$, but it
is also useful to consider larger rings like
${\mathbb{Z}}[\tfrac{1}{6},c_{4},c_{6}][(c_{6}^{2}-c_{4}^{3})^{-1}]$ that have
various universal properties in the theory of elliptic curves.) From this we
obtain a formal group $\widehat{C}$, which can be thought of as the part of
$C$ infinitesimally close to zero. It often happens that there is a spectrum
$E$ that corresponds to $\widehat{C}$ under the standard dictionary relating
formal groups to cohomology theories. Spectra arising in this way are called
_elliptic spectra_ [hoanst:esw]. The details are usually adjusted so that
$\pi_{*}(E)=k[u,u^{-1}]$ with $k=\pi_{0}(E)$ and $u\in\pi_{2}(E)$. In many
cases $E$ can be constructed using the Landweber Exact Functor Theorem, as in
(f).

The spectrum $TMF$ (standing for _topological modular forms_) “wants to be”
the universal example of an elliptic spectrum [dfhh:tmf]. It is not in fact an
elliptic spectrum, but it is close to being one, and it admits a canonical map
to every elliptic spectrum. If we let $MF_{*}$ denote the group of integral
modular forms as defined by number theorists (graded so that forms of weight
$k$ appear in degree $2k$) then we have

$\pi_{*}(TMF)[\tfrac{1}{6}]={\mathbb{Z}}[\tfrac{1}{6},c_{4},c_{6}][(c_{6}^{2}-c_{4}^{3})^{-1}]=MF_{*}[\tfrac{1}{6}].$

The significance of the number $6$ here is that $2$ and $3$ are the only
primes that can divide $|\operatorname{Aut}(C)|$, for any elliptic curve $C$.
If we do not invert $6$ then the homotopy groups $\pi_{*}(TMF)$ are completely
known, but different from $MF_{*}$ and too complex to describe here.

There is a dense network of partially understood interactions between elliptic
spectra, conformal nets [dohe:tmf], vertex operator algebras, chiral
differential operators [ch:wgv] and mathematical models of string theory
[stte:sft]. It seems likely that some central aspects of this picture remain
to be discovered.

Under (g) we discussed the algebraic $K$-theory spectrum $K(R)$ associated to
to a commutative ring $R$. The construction can be generalised to define
$K(R)$ when $R$ is a commutative ring spectrum; in particular, we can define
$K(kU)$. Rognes and Ausoni [auro:akt] have found evidence of a relationship
between $K(kU)$ and $TMF$, but many features of this remain obscure.

  * (l)

Let $R$ be a strictly commutative ring spectrum, so for any space $X$ we have
a ring $R^{0}(X)=[\Sigma^{\infty}X_{+},R]$, and a group of units
$R^{0}(X)^{\times}$. It can be shown that there is a spectrum $gl_{1}(R)$ such
that $R^{0}(X)^{\times}=[\Sigma^{\infty}X_{+},gl_{1}(R)]$ for all $X$. By
taking $X=S^{n}$ we see that $\pi_{n}(gl_{1}(R))=\pi_{n}(R)$ for $n>0$, but
$H_{*}(gl_{1}(R))$ is not closely related to $H_{*}(R)$, and many aspects of
the topology of $gl_{1}(R)$ are mysterious, even in the case $R=S^{0}$. For
any $(-1)$-connected spectrum $T$ one can build a strictly commutative ring
spectrum of homotopy type $\Sigma^{\infty}(\Omega^{\infty}T)_{+}$ (where
$\Omega^{\infty}T$ is the homotopy colimit of the spaces
$\Omega^{n}T({\mathbb{R}}^{n})$). Ring maps from this to $R$ then correspond
to maps $T\to gl_{1}(R)$ of spectra.

  * (m)

The notion of Bousfield localisation [bo:lspe, ra:lrc] provides an important
way to construct new spectra from old. The simplest kinds of Bousfield
localisations are the arithmetic ones: if $X$ is a spectrum and $p$ is a prime
number then there are spectra and maps
$X[\frac{1}{p}]\xleftarrow{i}X\xrightarrow{j}X_{(p)}$ such that $i$ induces an
isomorphism

$\pi_{*}(X)[\tfrac{1}{p}]=\pi_{*}(X)\otimes{\mathbb{Z}}[\tfrac{1}{p}]\to\pi_{*}(X[\tfrac{1}{p}])$

and $j$ induces an isomorphism

$\pi_{*}(X)_{(p)}=\pi_{*}(X)\otimes{\mathbb{Z}}_{(p)}\to\pi_{*}(X_{(p)}).$

Similarly, there is a map $X\to X{\mathbb{Q}}$ inducing
$\pi_{*}(X)\otimes{\mathbb{Q}}\simeq\pi_{*}(X{\mathbb{Q}})$. These properties
characterise $X[\frac{1}{p}]$, $X_{(p)}$ and $X{\mathbb{Q}}$ up to canonical
homotopy equivalence. Along similar lines, there is a $p$-adic completion map
$X\to X^{\wedge}_{p}$. If each homotopy group $\pi_{k}(X)$ is finitely
generated, then we just have
$\pi_{k}(X^{\wedge}_{p})=\pi_{k}(X)^{\wedge}_{p}=\pi_{k}(X)\otimes{\mathbb{Z}}_{p}$.
In the infinitely generated case the picture is more complicated, but still
well-understood.

Next, for any spectrum $E$ there is a functor
$L_{E}\colon{\mathcal{B}}\to{\mathcal{B}}$ and a natural map $i\colon X\to
L_{E}X$ characterised as follows: the induced map $E_{*}i\colon E_{*}X\to
E_{*}L_{E}X$ is an isomorphism, and if $f\colon X\to Y$ is such that $E_{*}f$
is an isomorphism, then there is a unique map $g\colon Y\to L_{E}X$ with
$gf=i$. The spectrum $L_{E}X$ is called the _Bousfield localisation_ of $X$
with respect to $E$, and we say that $X$ is _$E$ -local_ if the map $X\to
L_{E}X$ is a homotopy equivalence. The slogan is that the category
${\mathcal{B}}_{E}$ of $E$-local spectra is the part of stable homotopy theory
that is visible to $E$. Apart from the arithmetic completions and
localisations, the most important cases are the _chromatic localisations_
$L_{E(p,n)}$ and $L_{K(p,n)}$, which have been studied intensively [ra:lrc,
host:mkl]. It is a key point here that $E(p,n)$ and $K(p,n)$ have nontrivial
homotopy groups in infinitely many negative degrees. If the homotopy groups of
$E$ and $X$ are bounded below, it turns out that $L_{E}X$ is just an
arithmetic localisation of $X$, which is less interesting.

  * (n)

Under (b) we mentioned the spectra $X(n)$ and $MU=X(\infty)$. If we localise
at a prime $p$ (as discussed in (m)) then these can be split into much smaller
pieces. For example, we have $\pi_{*}MU={\mathbb{Z}}[x_{1},x_{2},\dotsc]$ with
$|x_{i}|=2i$, but $MU_{(p)}$ is equivalent to a coproduct of suspended copies
of a spectrum called $BP$, where
$\pi_{*}BP={\mathbb{Z}}_{(p)}[v_{1},v_{2},\dotsc]$ with $|v_{k}|=2(p^{k}-1)$.
This is called the _Brown-Peterson spectrum_. There are also spectra called
$T(n)$ that have the same relation to $X(p^{n})$ as $BP$ does to $MU$. This
whole story is most naturally understood in terms of formal groups and
$p$-typical curves. The spectra $T(n)$ play an important role in a number of
places, including the proof of the Nilpotence Theorem, the work of Mahowald,
Ravenel and Shick on the Telescope Conjecture [marash:tls], and Ravenel’s
methods for computation of stable homotopy groups of spheres [ra:ccs].

  * (o)

The theory of surgery aims to understand compact smooth manifolds by cutting
them into simpler pieces and reassembling the pieces [mami:css]. A key
ingredient is as follows: if we have an $n$-dimensional manifold with an
embedded copy of $S^{i-1}\times B^{j}$ (where $i+j=n+1$), then we can remove
the interior to leave a manifold with boundary $S^{i-1}\times S^{j-1}$, then
glue on a copy of $B^{i}\times S^{j-1}$ to obtain a new closed manifold
$M^{\prime}$. If the original copy of $S^{i-1}\times B^{j}$ is chosen
appropriately, then the cohomology of $M^{\prime}$ will be smaller than that
of $M$. By iterating this process, we hope to convert $M$ to $S^{n}$ (then we
can reverse the steps to obtain a convenient description of $M$). There are
various obstructions to completing this process (and similar processes for
related problems), and it turns out that these can be encoded as problems in
stable homotopy theory. Cobordism spectra such as $MSO$ play a role, as do the
spectra $kO$ and $gl_{1}(S^{0})$. When the dimension $n$ is even and $M$ is
oriented, the multiplication map $H^{n/2}(M)\otimes
H^{n/2}(M)\xrightarrow{}H^{n}(M)={\mathbb{Z}}$ gives a bilinear form on
$H^{n/2}(M)$, which is symmetric or antisymmetric depending on the parity of
$n/2$. Because of this, it turns out that we need to consider a kind of
$K$-theory of abelian groups equipped with a bilinear form. This is known as
$L$-theory. There are various different versions, depending on details that we
have skipped over. One of them gives a strictly commutative ring spectrum $L$
with $\pi_{*}(L)={\mathbb{Z}}[x,y]/(2x,x^{2})$, where $|x|=1$ and $|y|=4$ (so
for $k\geq 0$ we have $\pi_{4k}(L)={\mathbb{Z}}.y^{k}$ and
$\pi_{4k+1}(L)=({\mathbb{Z}}/2).y^{k}x$). There is a canonical map
$\sigma\colon MSO\to L$ of strictly commutative ring spectra [lamc:mpq]. For
an oriented manifold $M$ of dimension $4k$, the cobordism class $[M]$ gives an
element of $\pi_{4k}(MSO)$, so we must have $\sigma_{*}([M])=d.y^{k}$ for some
integer $d$. This integer is just the signature of the symmetric bilinear form
on $H^{2k}(M;{\mathbb{R}})$. A similar description can be given in dimension
$4k+1$.

## References


xspace upgreek