Let $X$ be a connected pointed topological space equipped with two different actions of $E\_\infty$-operad. Each action provides a collection of deloopings $X\_i$, where $X\_0 = X$ and $\Omega X\_i$ is homotopy equivalent to $X\_{i-1}$, so there are two $\Omega$-spectra having $X$ as a zeroth space. Can these spectra have different cohomology or different cohomology operations? I would like an explicit example or just a better reason than "why not".


I don't have any applications for such spectra. This question is motivated by thinking about D. Rector's paper [Loop Structures on the Homotopy Type of $S^3$](http://link.springer.com/chapter/10.1007%2FBFb0060900), which produces uncountably many different deloopings of three-dimensional sphere. They can be distinguished using cohomology operations.



Homologically distinct infinite loop structures on a space
There are easier ways to distinguish connective spectra with the same underlying space, if that's all you want to do. Like spaces, spectra have a theory of Postnikov towers, and in the same way that the Postnikov tower of a (simply connected, say) space $X$ measures the extent to which it differs from the product


$$\prod\_n B^n \pi\_n(X)$$


of Eilenberg-MacLane spaces, the Postnikov tower of a connective spectrum $E$ measures the extent to which it differs from the product


$$\prod\_n B^n \pi\_n(E)$$


of Eilenberg-MacLane spectra. 


Suppose $E$ is a spectrum with exactly two nontrivial homotopy groups $\pi\_n(E), \pi\_m(E), n < m$. Then $E$ naturally fits into a fiber sequence of spectra


$$B^m \pi\_m(E) \to E \to B^n \pi\_n(E)$$


and such fiber sequences are classified by a $k$-invariant given by a homotopy class of maps $B^n \pi\_n(E) \to B^{m+1} \pi\_m(E)$ of spectra, or equivalently by a stable cohomology operation


$$H^{\bullet + n}(-, \pi\_n(E)) \to H^{\bullet+m+1}(-, \pi\_m(E)).$$


If $n \ge 1$, then taking underlying spaces produces from the above fiber sequence of spectra a fiber sequence of connected spaces


$$B^m \pi\_m(E) \to E \to B^n \pi\_n(E)$$


which is a principal $B^m \pi\_m(E)$-bundle, and in particular which is classified by a $k$-invariant given by a homotopy class of maps $B^n \pi\_n(E) \to B^{m+1} \pi\_m(E)$ of spaces, or equivalently by an unstable cohomology operation


$$H^n(-, \pi\_n(E)) \to H^{m+1}(-, \pi\_m(E)).$$


In particular, we can find nontrivial infinite loop space structures on the space $B^n \pi\_n(E) \times B^m \pi\_m(E)$ by finding stable cohomology operations whose corresponding unstable cohomology operation in some degree is trivial.


*Example.* The simplest nontrivial case where the underlying space is connected occurs when we pick the $k$-invariant to be the second Steenrod square


$$\text{Sq}^2 : B \mathbb{Z}\_2 \to B^3 \mathbb{Z}\_2$$


(taking $\text{Sq}^1$ just gives $\mathbb{Z}\_4$ as a nontrivial extension of $\mathbb{Z}\_2$ by $\mathbb{Z}\_2$). This gives the unique nontrivial infinite loop space structure on the space $B \mathbb{Z}\_2 \times B^2 \mathbb{Z}\_2$, which arises naturally as the $2$-truncation of the connected component of the identity of the sphere spectrum; in particular, we know that the underlying space is $B \mathbb{Z}\_2 \times B^2 \mathbb{Z}\_2$ because $\text{Sq}^2$ as an unstable cohomology operation 


$$H^1(-, \mathbb{Z}\_2) \to H^3(-, \mathbb{Z}\_2)$$


vanishes. Equivalently, $\text{Sq}^2$ is nontrivial as a map of spectra but trivial as a map of spaces. 


The product spectrum structure on $B \mathbb{Z}\_2 \times B^2 \mathbb{Z}\_2$ gives the cohomology theory which is


$$H^1(X, \mathbb{Z}\_2) \times H^2(X, \mathbb{Z}\_2)$$


in degree $0$, with the product abelian group structure. By comparison, the nontrivial spectrum structure gives the cohomology theory which in degree $0$ has the same underlying set, but with the following modified abelian group structure, where $(w\_1, w\_2) \in H^1(X, \mathbb{Z}\_2) \times H^2(X, \mathbb{Z}\_2)$: 


$$(w\_1, w\_2) + (w\_1', w\_2') = (w\_1 + w\_1', w\_2 + w\_2' + w\_1 \cup w\_1').$$


This group is in general a nontrivial extension of $H^1(X, \mathbb{Z}\_2)$ by $H^2(X, \mathbb{Z}\_2)$; it arises as a kind of super Brauer group of $X$. The multiplication can be thought of as the cup product on the invertible elements $1 + w\_1 + w\_2 \in H^{\bullet}(X, \mathbb{Z}\_2) / H^{\bullet + 3}(X, \mathbb{Z}\_2)$ of a truncated version of the cohomology ring over $\mathbb{Z}\_2$.


