nLab Dyer-Lashof operation

Idea

An $n$-fold iterated loop space $\Omega^n X$ canonically carries the structure of an $E_n$-algebra (an algebra over the little n-disk operad $\mathcal{C}_n$) with $Sym(k)$-equivariant structure maps

$$\mathcal{C}_n(k) \times_{Sym(k)} \big( \Omega^n X \big)^k \overset{\mu_k}{\longrightarrow} \Omega^n X \,.$$

Passing to ordinary homology of the loop space with coefficients in a finite field, $H_\bullet\big( \Omega^n X; \mathbb{F}_p \big)$, the Dyer-Lashof operations are essentially the pushforward/images in homology under the binary operation $\mu_2$

$$H_\bullet\big( \mathcal{C}_n(2) \times_{Sym(2)} (\Omega^n X)^2 ; \mathbb{F}_p \big) \overset { (\mu_2)_\ast } {\longrightarrow} H_\bullet\big( \Omega^n X; \mathbb{F}_p \big) \mathrlap{\,.}$$

or rather, this map precomposed with

$$\begin{array}{ccc} H_i\big( \mathcal{C}_n(2)/Sym(2) ; \mathbb{F}_p \big) \times H_q\big( \Omega^n X ; \mathbb{F}_p \big) & \overset {\phantom{-} \boxtimes \phantom{-}} {\longrightarrow} & H_{i + 2 q}\big( \mathcal{C}_n(2) \times_{Sym(2)} (\Omega^n X)^2 ; \mathbb{F}_p \big) \\ (e_i, x) &\mapsto& e_i \boxtimes (x \otimes x) \mathrlap{\,.} \end{array}$$

Mote concretely:

$$\begin{aligned} \mathcal{C}_n(2) & \simeq Conf_2\big(\mathbb{R}^n\big) \\ & \simeq S^{n-1} \end{aligned}$$

is equivalently the configuration space of 2 points in $\mathbb{R}^n$, which in turn is equivalently the $(n-1)$-sphere of directions between these two points. Therefore

$$\mathcal{C}_n(2) \big/ Sym(2) \simeq \mathbb{R}P^{n-1}$$

is equivalently the real projective space.

For $p=2$, the ordinary homology of this space is (cf. there and use the universal coefficient theorem):

$$H_i\big( \mathbb{R}P^{n-1}; \mathbb{F}_2 \big) \simeq \left\{ \begin{array}{ll} \mathbb{F}_2 & \;\text{for}\; 0 \leq i \leq n-1 \\ 0 & \text{otherwise.} \end{array} \right.$$

Therefore there is a unique homology generator $e_i$ in each degree up to $n-1$.

Finally, the Dyer-Lashof operations $Q_i$ at the even prime are the above homology maps indexed by these generators:

$$\begin{array}{ccc} H_q\big( \Omega^n X ; \mathbb{F}_2 \big) &\overset{\phantom{-}Q_i\phantom{-}}{\longrightarrow}& H_{i + 2q}\big( \Omega^n X ; \mathbb{F}_2 \big) \\ x &\mapsto& (\mu_2)_\ast\big( e_i \boxtimes (x \otimes x) \big) \mathrlap{\,.} \end{array}$$

Related entries

• power operation

• homology ofiterated loop spaces

References

The original articles:

• Tatsuji Kudo, Shôrô Araki: Topology of $H_n$-Spaces and $H$-Squaring Operations, Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics 10 2 (1956) 85–120 [doi:10.2206/kyushumfs.10.85]

• William Browder: Homology operations and loop spaces, Illinois J. Math. 4 3 (1960) 347–357 [http://doi:10.1215/ijm/1255456051]

• Eldon Dyer, Richard Lashof: Homology of Iterated Loop Spaces, American Journal of Mathematics 84 1 (1962) 35–88 [doi:10.2307/2372804, jstor:2372804]

Further early discussion:

• Goro Nishida: Cohomology operations in iterated loop spaces, Proc. Japan Acad. 44 3 (1968) 104–109 [doi:10.3792/pja/1195521291]

The modern reformulation via operads is hinted at in

• Peter May; §5 in: The geometry of iterated loop spaces, Lecture Notes in Mathematics 271, Springer (1972) [doi:10.1007/BFb0067491, scan: pdf, retyped: pdf]

and expanded on in:

• Frederick R. Cohen, Thomas J. Lada, J. Peter May: The Homology of Iterated Loop Spaces, Lecture Notes in Mathematics 533, Springer (1976) [doi:10.1007/BFb0080464]

Review:

• Guozhen Wang: Dyer-Lashof Operations, talk slides (2018) [pdf]

• Tyler Lawson: $E_n$-ring spectra and Dyer-Lashof operations [arXiv:2002.03889]