I read that one of the main goals of utilization simplicial methods is to prove that a space is a loop space. On the other hand where lies the main importance to recognize topological spaces as loop spaces? Surely, if a space is a loop space then its connected components obtain a magma structure via concatenation because if $X$ is homotopy equivalent to a loop space $\Omega(Y)$ then $\pi_0(X)= \pi_1(Y)$ and thus there is a map $*: \pi_0(X) \times \pi_0(X) \to \pi_0(X)$.

Is this the only motivation behind the study of loop spaces? Or what other interesting aspects can be gained from studying loop spaces from the viewpoint of homotopy theory?

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    $\begingroup$ The rational cohomology of an infinite loop space is very nice: it is generated as a free graded commutative algebra by a basis for the rational cohomology of the associated spectrum. Why is this useful? Here’s one beautiful use of it. The Mumford conjecture is about the rational cohomology of the mapping class group, and it turns out the classifying space of the mapping class group is an infinite loop space. So figure out what the associated spectrum is and bam you’ve solved a huge problem just by knowing cohomology of loop spaces. $\endgroup$ Jul 12 '20 at 4:57
  • $\begingroup$ You might be interested in math.stackexchange.com/q/164118/10014 $\endgroup$ Jul 12 '20 at 6:36
  • $\begingroup$ I should say the classifying space of the mapping class group is homology equivalent to an infinite loop space, it can’t be an infinite loop space because the mapping class group is not abelian. $\endgroup$ Jul 12 '20 at 15:52
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    $\begingroup$ Being equivalent to a loop space is much stronger than \pi_0 being a group. By considering loops parametrized by [0, t] for all t\geq 0, you see that a loop space is a deformation retract of a (strictly unital and associative) topological monoid. But a strict algebra structure is not preserved by homotopy equivalence, so topological monoids are not a correct notion of 'homotopical associative monoid.' The definition of A-infinity algebra, which is the homotopically correct intrinsic notion of associative monoids, takes a cue from the recognition of algebraic structures present in loop spaces. $\endgroup$ Jul 13 '20 at 19:11
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    $\begingroup$ @crispr What I meant is that, while the magma structure on $\pi_0(X)$ in the question (so not an arbitrary one) is a group when $X$ is a loop space (which is already stronger than what OP said), we can even lift the multiplication on $X$ which is associative and unital "up to coherent homotopy" (whatever it means), and the existence of such a multiplication on $X$, in turn, characterizes loop spaces. This is the so-called 'recognition principle of the loop spaces' which probably OP had in mind. $\endgroup$ Jul 19 '20 at 1:54

There are several useful points in the comments, but I want to go beyond them and try to give a more comprehensive answer, so this question doesn't linger unanswered. Some great sources are May's Geometry of Iterated Loop Spaces (GILS) and A Concise Course in Algebraic Topology (CCAT). As the OP points out, if $X$ is a loop space, then concatenation makes $\pi_0(X)$ a group.

Is this the only motivation behind the study of loop spaces? Or what other interesting aspects can be gained from studying loop spaces from viewpoint of homotopy theory?

No, there is much more to loop spaces than the observation that concatenation yields a group structure on $\pi_0(X)$. First of all, motivation is easy. Since homotopy theory is intimately tied to paths, loop spaces themselves are fundamental objects, e.g., because they allow you to shift dimension as $\pi_i(\Omega X)\cong \pi_{i+1}(X)$. The operation of taking loops is connected to the suspension operation as follows. Let $X$ and $Y$ denote based spaces, $F(X,Y)$ denote the space of based maps between them (so $\Omega X = F(S^1,X)$), and $\Sigma X = X\wedge S^1$ denote the suspension of $X$. The usual hom-tensor adjunction tells you that $F(\Sigma X,Y) \cong F(X,\Omega Y)$. Taking $\pi_0$, we have $[\Sigma X,Y] \cong [X,\Omega Y]$, and composition of loops turns this set into a group. Hence, $\Omega Y$ is a cogroup object in the homotopy category of pointed spaces, and this is used in Hovey's book to set up the much more general homotopy theory encoded by model categories.

Since the theory of $\Omega$-spectra starts with a sequence of based spaces $T_n$ and weak equivalences $T_n\to \Omega T_{n+1}$, loop spaces are also foundational to stable homotopy theory. They pop up in the long exact sequences induced by fiber and cofiber sequences, that allow us to compute things in stable homotopy theory. They pop up in Postnikov towers and Brown representability (since $K(A,n) = \Omega K(A,n+1)$). One way to prove Bott periodicity is to study the homotopy equivalence of $H$-spaces $\beta: BU \times \mathbb{Z} \to \Omega^2(BU\times \mathbb{Z})$. So, there's plenty to motivate the study of loop spaces. Let's say more about "interesting aspects."

The introduction to Adams' book Infinite Loop Spaces mentions work of Morse and Serre computing the number of geodesics on a Riemannian manifold using loop spaces, work of Serre on $H$-spaces and the Pontryagin product on $H_*(X)$, and the development of the Leray-Serre spectral sequence and its resulting homology calculations. Loop spaces give us more to compute with, and double loop spaces, $n$-fold loop spaces, and infinite loop spaces, give us even more.

As has been pointed out in the comments, the recognition principle says that $n$-fold loop spaces $\Omega^n Y$ are (up to homotopy) the same thing as $E_n$-algebras. For $n=1$, these are the same as $A_\infty$-spaces, as discussed in the preface to GILS. The Moore path space trick Naruki mentioned (parametrizing loops by $[0,t)$) gives a model for a strictly associative and unital topological monoid of loops, and the usual loop space is a deformation-retract, which is one way to understand the $A_\infty$-space structure. It's easy to show that $\Omega Y$ is a grouplike homotopy associative $H$-space, but the $A_\infty$-structure is better.

As pointed out in the link Najib provided, the $i^{th}$ stable homotopy group of $X$ is equal to $\pi_{i+k} \Sigma^{k} X = \pi_i \Omega^k \Sigma^k X$ for sufficiently large $k$, so spaces of the form $\Omega^k \Sigma^k X$ for $1\leq k \leq \infty$ contain a tremendous amount of information about $X$. As pointed out in the preface to GILS, this leads you naturally to the James construction and to Dyer-Lashof operations, which are essential for understanding the algebraic structure of the (co)homology of $X$, for an understanding of power operations, and for computations in the Adams spectral sequence. In GILS, May finds geometric approximations to these spaces, and descriptions of $H_*(\Omega^n \Sigma^n X)$ as functors of $H_*(X)$. The resulting understanding of Dyer-Lashof operations is the foundation upon which much computational work has been done, as wonderfully summarized in an article of Tyler Lawson.

  • $\begingroup$ Thank you a lot for your great answer. One aspect I would like to understand deeper: You wrote It's easy to show that $\Omega Y$ is a grouplike homotopy associative $H$-space, but the $A_{\infty}$-structure is better. Could you explain a bit more precisely what you mean by "better"? $\endgroup$ Aug 9 '20 at 19:55
  • $\begingroup$ Sure. To be "homotopy associative" means associative in the homotopy category. That is, the associativity diagram (the two ways to get from $X\wedge X\wedge X$ to $X$) commutes up to homotopy. To be "associative" means it commutes on the nose (that the maps are literally equal). To be $A_\infty$ means it commutes up to infinitely coherent homotopy, e.g., a homotopy for the associativity diagram, plus homotopies between such homotopies, plus homotopies between those, etc. Way more structure than just the lowest level homotopy. More data/structure = better calculations. $\endgroup$ Aug 9 '20 at 20:33

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