Why spectra, and what stabilisation actually buys you

The Freudenthal suspension theorem says that for an $n$-connected space $X$, the suspension map

$$\pi_k(X) \longrightarrow \pi_{k+1}(\Sigma X)$$

is an isomorphism for $k < 2n + 1$ and surjective when $k = 2n + 1$. Iterate it and the homotopy groups stop moving. What is left is the stable homotopy group

$$\pi_k^s(X) = \operatorname{colim}_{n \to \infty} \pi_{k+n}(\Sigma^n X).$$

This is the whole trick, and it is worth being precise about what it costs.

What you gain

Unstably, $[X, Y]$ is a set — pointed, but a set. After stabilisation it becomes an abelian group, and the category of spectra is triangulated: cofibre sequences and fibre sequences agree, so a great deal of homological algebra transfers verbatim. The smash product makes it symmetric monoidal, with the sphere spectrum $\mathbb{S}$ as unit. Suddenly one can say “module over a ring spectrum” and mean it.

Concretely: the Adams spectral sequence

$$E_2^{s,t} = \operatorname{Ext}_{\mathcal{A}}^{s,t}(H^*(Y; \mathbb{F}_p),\, \mathbb{F}_p) \Longrightarrow \pi_{t-s}(Y)^{\wedge}_p$$

computes something genuinely hard — the $p$-completed stable homotopy of $Y$ — from something merely difficult, namely $\operatorname{Ext}$ over the Steenrod algebra $\mathcal{A}$.

What you lose

Stabilisation is not free. It forgets everything that is not visible after enough suspensions: the Whitehead product dies, and with it much of the unstable information about how $X$ is glued together. $\pi_*^s(S^0)$ is still, famously, not fully known.

Why a physicist should care

Anomalies in quantum field theory are classified by bordism groups, and bordism groups are homotopy groups of Thom spectra — $\Omega_n^{SO} \cong \pi_n(MSO)$. The invariants that a physical theory can actually detect are stable ones, which is a statement about spectra whether or not one chooses to say it that way.

That is roughly the thread I keep pulling on: if the observables are stable invariants, the right home for the question is the stable homotopy category.