17

This construction is found in Hatcher if I recall correctly. Let $p: EG \to BG$ be a quotient of a contractible space $EG$ by a free action of $G$. This gives rise to a fibration $G \hookrightarrow EG \twoheadrightarrow BG$. We also have the pathspace fibration of $BG$: Let $P(BG) = \{ \gamma : I \to BG : \gamma(0) = * \}$, then we have a fibration $\pi: P(...


12

Observe that any map $\mathbb{CP}^{n} \rightarrow \mathbb{CP}^{\infty}$ can be pushed down to the $2n$-skeleton of $\mathbb{CP}^{\infty}$, due to cellular approximation. This establishes a bijection between homotopy classes of maps $[\mathbb{CP}^{n}, \mathbb{CP}^{\infty}] \simeq [\mathbb{CP}^{n}, \mathbb{CP}^{m}]$ for all $m \geq n$. The left hand side ...


12

In general for a group $G$, there exists a contractible free $G$-space $EG$ whose quotient $BG=EG/G$ is a classifying space for $G$. The construction of $EG$ is not unique and may be carried out functorially using, say, a bar construction or Milnor infinite join construction. Whatever the method adopted, the space $EG$ exists and is unique up to a $G$-...


11

The homotopy theoretic proof is as follows: Let $E \longrightarrow \Sigma$ be a principal $G$-bundle over a surface $\Sigma$. Such a bundle is determined by a homotopy class $[f_E] \in [\Sigma, BG]$ by classifying space theory. Since $G$ is simply connected (and presumably connected), the classifying space $BG$ is $2$-connected (i.e. connected, simply ...


11

The functor $B : \mathbf{TopGrp} \to \operatorname{Ho} \mathbf{Top}_*$ is not fully faithful. Indeed, it does not even preserve non-isomorphy: the multiplicative group $\mathbb{R}^\times$ and the discrete group $\mathbb{Z} / 2 \mathbb{Z}$ have the same classifying space (namely, $\mathbb{R P}^\infty$), so if the functor were fully faithful, $\mathbb{R}^\...


11

For any topological group $G$, $EG \to BG$ is a principal $G$-bundle, so by the long exact sequence in homotopy, we have the long exact sequence $$\dots \to \pi_{n+1}(BG) \to \pi_n(G) \to \pi_n(EG) \to \pi_n(BG) \to \pi_{n-1}(G) \to \dots$$ As $EG$ is weakly contractible, $\pi_n(EG) = 0$ for $n > 0$, so we see that $\pi_{n+1}(BG) \cong \pi_n(G)$. In ...


10

The correct and invariant statement is that the classifying space functor is an equivalence of $(\infty, 1)$-categories between grouplike $E_1$ spaces (this is a homotopically invariant version of "topological group") and pointed connected spaces (and every time I say "spaces" I mean "weak homotopy types"). I believe it's also true that every grouplike $E_1$ ...


10

For a very simple example, consider the poset $\{a,b,c,d,e,f\}$ with $a,b\leq c,d\leq e,f$. The classifying space of this poset is homeomorphic to $S^2$ (you can explicitly list out all the nondegenerate simplices in its nerve and draw a picture of them), which has plenty of higher homotopy groups. More generally, in fact, every simplicial complex is ...


9

I'll try to fill in some of the missing details, hopefully this will be enough. Let the unoriented Grassmanian be $X = \widetilde{\mathrm{Gr}}(k, \mathbb{R}^n) \cong SO(n) / (SO(k) \times SO(n-k))$. Assume $0 < k < n$ (otherwise there's not much to prove). There is thus a fiber bundle $SO(n) \to X$, with fiber $SO(k) \times SO(n-k)$. Since $SO(n)$ is ...


8

The title and the body seem to be asking very different questions, so I'll answer them separately. Title question: Not quite. While it is true that every generalized cohomology theory is represented by a spectrum and conversely that every spectrum represents a generalized cohomology theory, maps between spectra are richer than maps between generalized ...


8

You can construct such a classifying space as a mapping telescope. Let $f:S^1\to S^1$ be a degree $2$ map and let $T$ be the telescope of the sequence $S^1\to S^1\to S^1\to\dots$ where the maps are all $f$. The homotopy groups of $T$ will then be the colimits of the induced sequence of maps on the homotopy groups of $S^1$. This means $\pi_n(T)=0$ for $n\...


7

First, let me make the weaker claim that $\mathbb{HP}^{\infty}$ is not naturally an H-space. The natural H-space structures on $\mathbb{RP}^{\infty}$ resp. $\mathbb{CP}^{\infty}$ come from the fact that they classify isomorphism classes of real resp. complex line bundles, which naturally have group structures given by taking the tensor product. This no ...


7

In general, $$[X, K(G, n)] \cong H^n(X; G)$$ where $K(G, n)$ is an Eilenbeg-MacLane space (see Theorem $4.57$ of Hatcher's Algebraic Topology). In particular, if $G$ is equipped with the discrete topology, then $BG$ is a $K(G, 1)$ so $$[X, BG] \cong [X, K(G, 1)] \cong H^1(X; G).$$ On the other hand, for an abelian group $G$, $$H^1(X; G) \cong \...


7

For your first question, there are many reasonable constructions of these. One way is via linearization of spheres (see the section at https://ncatlab.org/nlab/show/Eilenberg-Mac+Lane+space). Another way to think about it is through the Dold-Kan correspondence. In this lens, an Eilenberg-MacLane space is just the space associated to the chain complex with ...


6

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. (It's also called the internal nerve.) The geometric realization $BM=|N_\bullet(M)|$ of this simplicial space is the classifying space $BM$. (This includes the discrete case.) Unveiling the definitions, ...


6

If $G$ is a topological group, then there is a universal principal $G$-bundle $EG \to BG$ where $EG$ is weakly contractible. Using the long exact sequence in homotopy, we see that $$\dots \to \pi_{i+1}(EG) \to \pi_{i+1}(BG) \to \pi_i(G) \to \pi_i(EG) \to \dots$$ As $EG$ is weakly contractible, $\pi_{i+1}(EG) = 0$ and $\pi_i(EG) = 0$, so $\pi_{i+1}(BG) = \...


5

There are Chern classes defined on K-theory groups with values in cyclic homology and Hochschild homology, which are the exact analogue of characteristic classes for general algebras —in fact, these can be defined using those. You can find this explained in Loday's book on cyclic homology, in Karoubi's book on the same subject, in Rosenberg's introduction ...


5

One can form $BM$ for any $A_\infty$-space and it's more or less the delooping ($A_\infty$-structure on a connected $H$-space $M$ is more or less the same thing as an equivalence $M\cong \Omega X$ for some $X$; the proof is more or less that $M\cong\Omega BM$). (AFAIR this can be generalized further but then there is a question of what properties do you ...


5

I just consider the case $h=3$, but the argument is completely general. From the classification of closed surfaces, we know that $N_3$ is the connected sum of three projective spaces, so that $N_3$ be the quotient of an hexagon $P$ by identifying its sides as indicated by the following figure: Now let $U, V \subset N_3$ be subspaces illustrated by the ...


5

The answer to both your questions is yes, and Qiaochu gave the basic idea. The base space is $BS_n$ and the fiber is $ES_n$. You can make this concrete (very analogous to Grassmannians) by using the model $BS_n \equiv C_n(\mathbb R^\infty) / S_n$ and $ES_n = C_n(\mathbb R^\infty)$ where $C_n$ indicates the configuration space of $n$ labelled points in $\...


5

Every monoid is a category with one object. Every small category has a classifying space, defined as the geometric realization of the nerve. The classifying space of a monoid is (by definition) the classifying space of the corresponding category. Explicitly, if $M=(X,\cdot,1)$ is a monoid, its classifying space $BM$ is a CW-complex which has an $n$-cell ...


5

It's easier to think about this the other way around. By definition, $C'_n$ is the set of pairs $(c,t)$ where $c \in C_n$ is a configuration, $t \ge 0$ is a real number, and the configuration $c$ is in $(0,t) \times \mathbb{R}^{n-1}$. Then there is a homotopy equivalence $\psi : C'_n \to C_n$ that simply forgets the parameter $t$ (ie. $\psi(c,t) = c$). This ...


5

Up to homotopy, the answer is if and only if $G$ has the additional structure of an $E_2$ space. (This is exactly the structure that a double loop space has; we need this since a topological group has a classifying space, so if $BG$ has a classifying space then $G$ is a double loop space.) This is a certain higher categorical version of abelianness. If $G$ ...


5

The Baumslag-Solitar group $G=\mathbf{Z}[1/2]\rtimes\mathbf{Z}$ acts properly (and freely) on a contractible space, namely the product $P$ of the hyperbolic plane and of a trivalent tree $T$, which can be chosen to be either the Bruhat-Tits tree of $\mathrm{SL}_2(\mathbf{Q}_2)$, or the Bass-Serre tree when $\mathbf{Z}[1/2]\rtimes\mathbf{Z}$ is view as an ...


5

A connected Lie group $G$ is diffeomorphic to $K\times\mathbb{R}^d$ for some $d$ where $K$ is a maximal compact subgroup of $G$. This was proved by Élie Cartan in the semisimple case, and independently by Malcev and Iwasawa in the general case; see this MathOverflow question. It follows that $G$ is homotopy equivalent to $K$ and therefore $BG$ is homotopy ...


5

Let $G$ be a topological group with classifying space $BG$. Suppose that $BG$ supports a group structure. Then there are homotopy equivalences $G\simeq \Omega BG\simeq\Omega^2B^2G$, so $G$ is a double loop space, and in particular supports a homotopy abelian H-space multiplication. However: Theorem: Let $X$ be a non contractible, connected, finite complex ...


4

Recall that group cohomology is not just about the trivial module, but is something you can compute for all $G$-modules. The corresponding thing on the topological side is to consider local systems on $K(G, 1)$ and their cohomology; in fact the category of local systems on $K(G, 1)$ is equivalent to the category of $G$-modules. Moreover, just as group ...


4

No. If unit octonions were $A_\infty$-space, $BS^7$ («$OP^\infty$») would be a space with cohomology ring $\mathbb Z[x]$, $\deg x=8$ (this follows from LHSS for Serre fibration $\text{pt}\to\Omega BS^7\cong S^7\to BS^7$) — which is impossible (see e.g. Corollary 4L.10 if Hatcher's «Algebraic topology»). P.S. In fact, there are no even just homotopy ...


4

Real resp. complex $n$-dimensional vector bundles are classified by homotopy classes of maps $M \to BO(n)$ resp. $M \to BU(n)$. Stable real resp. complex vector bundles are classified by homotopy classes (not stable homotopy classes) of maps $M \to BO$ resp. $M \to BU$, where $O$ is the stable orthogonal group and $U$ is the stable unitary group. This ...


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