37
votes

Accepted

### If $\Omega X \simeq \Omega Y$ then is $X \simeq Y$ for $X,Y$ simply connected?

To expand on my comment: As written (i.e. without requiring a map $f:X\to Y$), this is false in general. For an example, write $S^2$ as homogeneous space, $S^2=SU(2)/U(1)$. This exhibits $S^2$ as the ...

19
votes

Accepted

### CW complex of iterated loop spaces

By a result of Milnor, the space of maps from a finite CW complex to any CW complex is homotopy equivalent to a CW complex. This gives a general reason why spaces like $\Omega^k M$ have a CW structure....

17
votes

Accepted

### Homology of the free loop space of a Grassmanian

The complex Grassmannian $Gr(2,4)$ can be realized up to homotopy as the homotopy fiber of the map $BU(2) \times BU(2) \rightarrow BU(4)$ which corresponds to the Whitney sum of two complex rank 2 ...

17
votes

Accepted

### The free loop space of spheres

I am grateful to Tobias Barthel, who sent me the following paper of J. Aguadé:
"On the space of free loops of an odd sphere". Pub. Mat. UAB No 25, June 1981.
Aguadé proves the following theorem
...

16
votes

Accepted

### For which G is BLG weak homotopy equivalent to LBG?

[UPDATE: There were some mistakes in the first version. Here is a more careful account.]
I'll work everywhere with CGWH spaces, so I have a Cartesian closed category.
Note that $BLG$ is always path-...

16
votes

Accepted

### Homotopy type of continuous/smooth/analytic loop spaces?

Suppose $S$ and $M$ are smooth manifolds. For simplicity let us also suppose that $S$ is compact. Then the inclusion $C^\infty(S, M)\hookrightarrow C^0(S, M)$ is a weak equivalence. This is a "...

13
votes

Accepted

### When does $BG \to BA$ loop to a homomorphism?

If $G$ is a compact connected topological group and $A$ is a locally compact abelian topological group, then for any map $f:BG\to BA$ the looped map $\Omega f:\Omega BG\to \Omega BA$ is homotopically ...

13
votes

Accepted

### Homotopy orbits, spectra and infinite loop spaces

Both of these are false.
The first is close to true: if $S(n-1)$ is the unit sphere in $\Bbb R^{n}$ with its standard $O(n)$-action, then we can identify $S(n-1)$ with $O(n) / O(n-1)$ and so get the ...

12
votes

Accepted

### Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$

First note that Hatcher's exercise says "where direct limits mean mapping telescopes", so he is defining $\underset{\rightarrow}{\lim}$ to mean the telescope. I disapprove of that quite strongly. ...

11
votes

Accepted

### How are characteristic classes morphisms of infinite loop spaces? (if they are)

Your sequences are all arise in the following standard way. Suppose $x$ is an $(n-1)$--connected spectrum and let $X = \Omega^{\infty} x$. One always has a
fibration sequence
$$y \rightarrow x \...

11
votes

### How are characteristic classes morphisms of infinite loop spaces? (if they are)

Yes it is true. You have correctly interpreted the intended meaning of the phrase ``fibration of infinite loop spaces''. One early reference is chapter I of $E_{\infty}$ ring spaces and $E_{\infty}$ ...

11
votes

Accepted

### Is the homology of $\Omega^2\Sigma^2X$ free as a Gerstenhaber algebra?

Over $\mathbb{Z}_p$ it is not true that $H_*(\Omega^2\Sigma^2X)$ is the free Gerstenhaber algebra. Instead, Cohen proves that $H_*(\Omega^n\Sigma^nX)$ is a free object in a more elaborate category ...

10
votes

Accepted

### Stable homotopy groups of $QX$

The "Snaith splitting" gives the following spectrum level statement: for a pointed connected space $X$, there is a weak equivalence:
$$
\Sigma^\infty_+ (\Omega^\infty \Sigma^\infty X) \simeq \bigvee_{...

10
votes

Accepted

### Delooping the quotient space $SU/SU(n)$

I'll work with mod $2$ cohomology. Note that $H^*(BSU(2))$ is polynomial on $c_2$ (in degree $4$) and $H^*(BSU)$ is polynomial on $c_k$ for $k\geq 2$. Here $c_k$ has degree $2k$ and so $H^6(BSU)=\{0,...

9
votes

### Relative homology of free loop space with respect to constant loops

Let me tackle the case that $Q$ is $1$-connected, but not $2$-connected. Because $Q$ is $1$-connected we have that $\Lambda_0Q=\Lambda Q$, as all loops are contractible.
Two sequences are relevant ...

9
votes

Accepted

### Homology of the free loop space of generalized flag varieties

Serre proved that for any simply-connected $X$, if $X$ has finitely generated homology groups in each degree, then the loop space of $X$ has finitely generated homology groups in each degree. (...

9
votes

### Jones' theorem for non-simply-connected spaces?

Corollary 9.5 in the paper Rivera and Zeinalian - Cubical rigidification, the cobar construction, and the based loop space tells us the following:
For any pointed path connected space $(X,b)$ the co-...

8
votes

Accepted

### The free smooth path space on a manifold

Yes.
The technical details are in Yet More Smooth Mapping Spaces and Their Smoothly Local Properties, specifically in Section 5 which establishes that smooth manifolds are smoothly locally deformable ...

8
votes

Accepted

### Loop spaces motivation

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 ...

8
votes

### Integral homology of braid groups as a ring

Using the Hopf fibration you can show that $\Omega^2S^2\simeq\mathbb{Z}\times\Omega^2S^3$. Also $\pi_1(\Omega^2S^3)=\pi_3(S^3)=\mathbb{Z}$, so the Universal Coefficient Theorem gives $[\Omega^2S^3,S^...

8
votes

Accepted

### Is there a filtered splitting of product labelling spaces?

The answer to your first question is no. And this can be seen by homology considerations. Note that this equivalence induces an isomorphism of Hopf algebras
$$ H_*(C(\mathbb R; X \vee Y \vee (X\...

8
votes

Accepted

### How now to study operads in homotopy theory?

One of the most comprehensive references today is certainly:
B. Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups, Mathematical Surveys and Monographs 217. https://bookstore.ams.org/...

6
votes

### Integer homology of double loop space of odd-dimensional sphere

The calculation you want is described in detail in Joe Neisendorfer's book Algebraic Methods in Unstable Homotopy Theory, Cambridge University Press, 2010. In particular, the Eilenberg--Moore ...

6
votes

Accepted

### Integer homology of double loop space of odd-dimensional sphere

There are homology isomorphisms $K(Br,1)\to \Omega^2_0 S^2$ and $\Omega^2 S^3\to \Omega^2_0 S^2$, so you are really asking about the homology of the stable braid group $Br$ (the colimit of the natural ...

6
votes

Accepted

### Adjoint map of $\Gamma$-space prespectrum

It is proposition 1.4 in Segal's Categories and cohomology theories (a paper I love and I strongly encourage everyone interested in homotopy theory to read).

6
votes

### Is there a good way to understand the free loop space of a sphere?

I recently learned the following description of the topology of the loop space at the European Talbot on free loop spaces, and is probably what Ziller does in the paper in the answer of Lennart Meier. ...

6
votes

Accepted

### Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself?

I don't know what you would mean by the normal bundle of $M$ in $LM$, but the answer to one question is "no": When the $\mathbb Q[u]$-module $H^\ast_{S^1}(LM)$ is localized by inverting $u$, ...

5
votes

Accepted

### Sheafification of loop scheme/group

Despite Will Sawin's answer in comment, it was not so obvious for me at first glance, so I give a detailed (at least for $X$ affine) proof here.
Lemma. Let $A$ be a $K$-algebra and let $A\to B$ be ...

5
votes

Accepted

### Is there a fibration sequence of spectra $K\mathbb{F}_q\to KU\to KU$?

The infinite loop space/spectrum level statements were written down in
May, Quinn, Ray, Tornehave:
"$E_\infty$ Ring Spaces and $E_\infty$ Ring Spectra" (1977) http://www.math.uchicago.edu/~may/BOOKS/...

5
votes

Accepted

### Is $\Omega J_{p^n-1}S^2$ commutative up to homotopy?

This was answered in the affirmative by Brayton Gray in his paper Homotopy Commutativity and the EHP Sequence. Specifically he shows that for all $n$ the space $\Omega J_{p^s-1} S^{2n}$ is homotopy ...

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