at.algebraic topology – What is the homotopy category of the sphere spectrum?

Since the sphere spectrum is connective, we may view it as an $mathbb{E}_{infty}$-group in spaces, namely $QS^0$. This space, as any simplicial set, has a homotopy category $mathsf{Ho}(QS^0)$, and since $mathsf{Ho}$ is symmetric monoidal, $mathsf{Ho}(QS^0)$ acquires the structure of an $mathbb{E}_{infty}$-group in categories, making it into a “grouplike” symmetric monoidal category (more commonly called a symmetric groupoidal category or an abelian $2$-group).

Is there a known explicit description of the symmetric groupoidal category $mathsf{Ho}(mathbb{S})overset{mathrm{def}}{=}mathsf{Ho}(QS^0)$?

at.algebraic topology – What happens to a closed manifold to ensure it is homeomorphic to a torus $T^{n}$?

For dimensions $n geq 5$, the answer is yes. First, note that $M$ is homotopy equivalent to a torus since it must be a $K(mathbb{Z}^n,1)$. Second, Hsiang-Wall show in “On Homotopy Tori II” that in such dimensions, homotopy tori are in fact actual tori (i.e. homeomorphic to tori).

at.algebraic topology – Every homeomorphism isotopic to one with finitely many fixed points

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at.algebraic topology – Is there a concrete description of the nontorsion elements in the homotopy groups of spheres?

By Serre’s theorem, we know the only nontorsion parts of the homotopy groups of spheres occur as $pi_n(S^n)$ and $pi_{4n-1}(S^{2n})$. The first of these are trivial to describe, but the second have very interesting, symmetric incarnations, they are the generalised hopf fibrations, at least for $n=1,2,4$, associated to the real normed division algebras.

Are there a similar explicit descriptions for representatives of these higher nontorsion elements too?

Even if we don’t have explicit descriptions, do we know anything about the values of the hopf invariants associated to these maps?

at.algebraic topology – What is the pointed Borel construction of the $0$-sphere?

From what I understand, the Borel construction takes a $G$-space $X$ and produces a topological space $Xtimes_{G}mathbf{E}G$―the homotopy quotient $X/!!/G$ of $X$ by $G$ in the $infty$-category of spaces $mathcal{S}$―satisfying
One partiularly important example is given by $mathbf{B}G$, the Borel construction / homotopy quotient of the point: $mathbf{B}Gsimeqmathbf{E}Gtimes_G*$.

Moving to the pointed setting, one has the pointed Borel construction, which takes a pointed $G$-space $X$ and returns $mathbf{E}G_+wedge_{G}X$, the homotopy quotient of $X$ by $G$ in the $infty$-category of pointed spaces $mathcal{S}_*$. Concretely, it is given by
mathbf{E}G_+wedge_{G}X &overset{mathrm{def}}{=} frac{mathbf{E}G_+times_G X}{mathbf{E}Gtimes_G*},\
&cong frac{mathbf{E}G_+times_G X}{mathbf{B}G}.

Now, $mathbf{E}G_+wedge_G*cong*$, rather than $mathbf{B}G$. But while $*$ is the monoidal unit of $mathcal{S}$, it is not that of $mathcal{S}_*$, which is $S^{0}$. Hence it would be interesting to know:

Question. What is the pointed Borel construction $mathbf{E}G_+wedge_G S^0$ of the $0$-sphere? Is it related to $mathbf{B}G_+$?

at.algebraic topology – Necessary and sufficient conditions for the Lie group embedding $G supset J$ can be lifted to $G$’s covering space

Suppose the Lie group $G$ contains the Lie group $J$ as a subgroup, so
G supset J.

If $G$ has a nontrivial first homotopy group $pi_1(G) neq 0$.

If $G$ has a universal cover $widetilde{G}$, so $pi_1(widetilde{G}) = 0$.

Question: What are the necessary and sufficient conditions to derive that
widetilde{G} supset J quad (?)

is also true via a lifting?
& & widetilde{G}\
&nearrow & downarrow\
J & longrightarrow & G

My thought:

  1. Sufficient condition may be $pi_1(J)=0$, due to the property $pi_1(widetilde{G})=0$.

  2. Necessary condition may be that
    $pi_1(J) to pi_1({G})$ is a trivial group homomorphism, which maps all $pi_1(J)$ to 0,
    because this map
    can be decomposed by $pi_1(J) to pi_1(widetilde{G})=0$ and $pi_1(widetilde{G})=0 to pi_1({G})$.

  3. The above conditions are for universal covering $widetilde{G}$, what would be the conditions for the (non-universal) covering space $widetilde{G}$?

at.algebraic topology – Dimension of a CW complex with free abelian fundamental group

If I start with a finite CW complex $X$ with $pi_1(X)=mathbb{Z}^r$(free abelian group of rank $r$) where $rgeq 3.$ I want to claim the dimension of the CW complex is $geq 3.$

I started thinking by taking the example of the $r$ torus in my head but could not get proper proof.

at.algebraic topology – Equivariant line bundles and connections

Equivariant line bundle isomorphism classes are classified by the equivariant cohomology group $H^2_{P}(X;mathbb{Z})$ and let us take $P$ to be finite abelian and $X$ a finite dimensional CW-complex to simplify matters.

I was wondering how to understand these classes in terms of the connection $A$ of the associated $U(1)$-bundle. I know of some examples where it is not the curvature $F_{A}$ but the connection $A$ itself integrated over a subregion (or boundary of said subregion) fixed under the $P$-action to obtain some of these classes.

Is there a general procedure to describe any given class in terms of an integral of the connection $A$ or its curvature $F_{A}$ over some subregion of $X$ and how does this region relate to the $P$-action?

Thanks for reading

at.algebraic topology – Cannot follow a cohomology computation

I am trying to understand a proof from Toda’s paper Cohomology of classifying spaces. The step I am stuck on is at page 96. Here is the setup.

We work with cohomology with $mathbb{F}_2$ coefficients. Let $G$ be a compact Lie group, $Gammasubset G$ be a central subgroup which is isomorphic to either $S_1$ or $mathbb{Z}/2$, let $E_1:=H^*(B^2Gamma)otimes H^*(BG)$ with differential
$$d:=(muotimes 1)circ (1otimes thetaotimes 1)circ (1circ phi),$$

where $phi:H^*(BG)to H^*(BGamma)otimes H^*(BG)$ is the coaction map (i.e. the map induced by the multiplication $Gammatimes Gto G$, which is a group homomorphism as $Gamma$ is central), $mu$ is the multiplication map, and $theta$ is the $mathbb{F}_2$-linear map defined as follows: if
$$H^*(BGamma)=mathbb{F}_2(x_2),qquad H^*(BBGamma)=mathbb{F}_2(z_2,z_3,z_5,dots,z_{2^k+1},dots)$$ (where $z_2=0$ when $Gamma=S^1$), then
$$theta(x_2^{2^i})=z_{2^{i+1}+1}, qquad theta(x_2^j)=0 text{ if $j$ is not a power of $2$}.$$

Simply put, I don’t understand the reasoning at page 96, which from (4.2) deduces Theorem 4.1. Let me be more specific. Assume that $G=SU(n)$, $Gamma=S_1$, and let $q$ be the highest power of two dividing $n$.

At page 96, right before the statement of Theorem 4.1, it is claimed that

$$(*)qquad H^*(E_1,d)=H^*(Cotimes B,d)$$

where $C=mathbb{F}_2(z_3,dots,z_q)$ and $B={ain H^*(BG): d_i(a)=0 text{ for all $igeq q$}}$, where the $d_i$ are the components of $phi$: $phi(a)=sum x^iotimes d_i(a)$. The proof of $(*)$ is supposedly in the few sentences leading up to the statement of Theorem 4.1, but I can’t quite follow it.

at.algebraic topology – Induced map in homology for a map to a loop space

Suppose $Y$ is an $(n-1)$-connected space, $n>2$, so we have Hurewicz isomorphisms $pi_n(Y)cong H_n(Y)$ and $pi_{n-1}(Omega Y)cong H_{n-1}(Omega Y)$. Let a map $alphacolon XtoOmega Y$ be given. Naturally it induces a map $betacolon Xtimes S^1to Y$. I want to show the following diagram is commutative:
H_{n-1}(X) @>times(S^1)>> H_n(Xtimes S^1)\
@Valpha_*VV @Vbeta_*VV \
H_{n-1}(Omega Y) @<cong<< H_n(Y).

Since $Omega Y$ is path connected, we may homotope $alpha$ to a based map. Then $beta$ factors though the reduced suspension $Sigma X$. If $X=S^{n-1}$ is a sphere, the commutativity would then follow from tracking down the definition of $pi_n(Y)xrightarrow{cong}pi_{n-1}(Omega Y)$. However I don’t know how this helps for the general case.

One can also phrase the question in cohomology in the obvious way. (In particular the cross product $times(S^1)$ will be replaced by the slant product $/(S^1)$.)