Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

**2**

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**0**answers

113 views

### Abelian covers of compact Kahler manifolds

Let $X$ be a compact Kahler manifold and $A\subset H_1(X,\mathbb{Z})$ be a subgroup.
Corresponding to $A$ there is an abelian covering $X_A \to X$ with $Deck(X_A)=H_1(X,\mathbb{Z})/A$. For example if ...

**1**

vote

**1**answer

220 views

### Relations between characteristic classes of a group and the Stiefel-Whitney/Pontryagin classes

Let $X$ be a closed manifold and $BG$ be the classifying space of a group $G$
A map from $X$ to $BG$ induce a map from $H^*(BG,Z)$ to $H^*(X,Z)$ by pull back.
Let $GH^*(X,Z)$ be the subgroup of ...

**27**

votes

**0**answers

505 views

### Are there periodicity phenomena in manifold topology with odd period?

The study of $n$-manifolds has some well-known periodicities in $n$ with period a power of $2$:
$n \bmod 2$ is important. Poincaré duality implies that odd-dimensional compact oriented manifolds ...

**11**

votes

**0**answers

371 views

### Chiral categories versus braided monoidal categories

Let $X$ be a curve over $\mathbf{C}$. As I understand from the 2008 Talbot notes, a chiral category on $X$ consists of a crystal of categories on the Ran space $\mathrm{Ran}(X)$ (see these notes of ...

**12**

votes

**2**answers

346 views

### How to compute $\pi_{3}$ of $L(p,q)\# L(p',q')$?

Let $L(p,q)$ be a 3-dimensional lens space, and let $L(p',q')$ be another. Is there any known result concerning the 3rd homotopy group of the connected sum $L(p,q)\# L(p',q')$? If not, I am interested ...

**1**

vote

**0**answers

98 views

### Why is cellularization the fiber of nullification for slice cells?

I'm a bit confused about the nullification functors that come up when constructing the slice tower in HHR.
Let $\mathcal{A}$ be a set of compact objects in the $G$-equivariant stable homotopy ...

**16**

votes

**0**answers

231 views

### Elliptic $\infty$-line bundles over $B G$

Theorem 5.2 in Jacob Lurie's "Survey of Elliptic Cohomology" (pdf) states the equivalence of two maps
$$
B G \longrightarrow B \mathrm{GL}_1(A)
$$
for $A$ an $E_\infty$-ring carrying an oriented ...

**23**

votes

**2**answers

978 views

### What are the “correct” conventions for defining Clifford algebras?

I have three related questions about conventions for defining Clifford algebras.
1) Let $(V, q)$ be a quadratic vector space. Should the Clifford algebra $\text{Cliff}(V, q)$ have defining ...

**8**

votes

**1**answer

273 views

### Existence of certain “nondegenerate” function and manifold topology

Let $M$ be a smooth manifold without boundary, not necessarily compact.
Let $f$ be a real-valued smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local ...

**1**

vote

**0**answers

84 views

### Are there analogs of smooth partitions of unity and good open covers for PL-manifolds?

Smooth partitions of unity and differentiable good open covers are important technical tools in the realm of smooth manifolds.
Are there analogs of these tools for piecewise linear manifolds?
A PL ...

**2**

votes

**0**answers

110 views

### generalized Atiyah-Hirzebruch spectral sequence from Postnikov truncation

The Atiyah-Hirzebruch spectral sequence
\begin{equation*}E^2_{p,q}=H_p(B,h_q(F))\Rightarrow h_{p+q}(E),\end{equation*}
computes the generalized homology $h$ of a total space $E$ of a Serre fibration ...

**6**

votes

**0**answers

244 views

### Classifying space of the higher-structure diffeomorphism group

There is a higher extension of the classifying space $B \mathrm{Diff}$ of the diffeomorphism group implicit in the (infinity,n)-category of cobordisms with (X,zeta)-structure ...

**5**

votes

**2**answers

251 views

### Which groups are LERF?

A finitely generated group $G$ is called LERF if every finitely generated $H \leq G$ is closed in the profinite topology on $G$ (equivalently, there is a family of finite index subgroups of $G$ ...

**6**

votes

**2**answers

425 views

### Reference for a fact (?) on homeomorphic knot complements

Does somebody have a reference (or an argument why it should be true) for the following statement?
“Let $K$ and $K'$ be knots in $S^3$. If there is an orientation-preserving homeomorphism $h : S^3 ...

**2**

votes

**0**answers

127 views

### Marshall Hall's theorem for surface groups [closed]

Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$
Let $H \leq \Gamma_g$ ...

**3**

votes

**1**answer

299 views

### Doing some homological algebra in triangulated categories

It's well known that chain complexes are an abelian category, and in particular we can consider chain complexes of chain complexes, i.e. double complexes. Given a double complex $A^{\bullet\bullet} ...

**1**

vote

**1**answer

154 views

### Free action of symmetric groups

What type of compact manifolds, can be acted freely by symmetric group $S_{m}$ for some $m>2$?
Is there a compact manifold which can be act freely by all symmetric ...

**6**

votes

**1**answer

187 views

### A homological criterion for collapsibility?

On page 256 of Kirby and Siebenmann one finds the following lemma (its proof an "elementary exercise", so they only give a hint):
Taking $A$ to be a point and iterating this collapsing lemma, this ...

**18**

votes

**2**answers

780 views

### Teaching the fundamental group via everyday examples

This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...

**9**

votes

**0**answers

175 views

### Mixed Hodge structure on configuration spaces

Let $X$ be a smooth complex projective variety. Let $F(X,n)$ be the configuration space parametrizing $n$ distinct ordered points in $X$. The cohomology groups $H^k(F(X,n),\mathbf Q)$ carry a mixed ...

**30**

votes

**4**answers

2k views

### Why is it so hard to compute $\pi_n(S^n)$?

Of course it isn't really that hard - nowhere near as hard as $\pi_k(S^n)$ for $k>n$, for instance. The hardness that I'm referring to is based on the observation that apparently nobody knows how ...

**7**

votes

**1**answer

186 views

### cohomology of classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$.
Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$.
Let $\rho: ...

**2**

votes

**0**answers

153 views

### A possible generalization of the Borsuk Ulam theorem via action of symmetric groups

The symmetric group $S_{m}$ is equiped with the counting Har measure $\mu$ and the obvious sgn character. Assume that $S_{m}$ acts on $S^{n}$, $n\geq m-1$ and $f:S^{n}\to \mathbb{R}^{n}$ ...

**5**

votes

**1**answer

325 views

### Can an odd map be null homotopic?

Let $G$ be a compact Lie group with invariant measure $\mu$. An odd function is a continuous function, $\phi:G\to \mathbb{C}$, such that $\int_{G} \phi d\mu=0$. An odd map is a continuous map, $f:G\to ...

**9**

votes

**3**answers

384 views

### Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$

Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$.
Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$?
...

**5**

votes

**1**answer

334 views

### Associative Ring Spectra and Derived Completion

So, I was thinking before that this might have some nice, simple topos theoretic explanation, but Jacob disabused me of that notion. However, I'm still very interested in the following question:
Is ...

**0**

votes

**0**answers

77 views

### Normal Form of Homotopy Pure Braids?

It is well known that a pure braid has a normal form (also called the combed form). Namely, let $P_n$ be the set of pure braids of $n$ strands and let $d_i:P_n\to P_{n-1}$ be the $i$th "forgetting ...

**8**

votes

**1**answer

556 views

### Higher vector spaces

As far as I know there are different ways to categorify the notion of vector space/module. These appear (for example) when trying to find extended TQFTs. There are at least two ways (presented at ...

**10**

votes

**1**answer

450 views

### Is there any relationship between the Euler class and the Vandermonde determinant?

Several Wikipedia articles claim that the relationship between the Euler class $e(V)$ and the top Pontryagin class $p_k(V)$ of an oriented $2k$-dimensional real vector bundle $V$ corresponds, via the ...

**3**

votes

**0**answers

67 views

### Integral group rings on which stably free modules are free

Let $G$ be a torsion-free group and $ZG$ the integral group rings. Recall that a projective module $P$ over $ZG$ is stably free if there is an isomorphism $P \oplus ZG^n \cong ZG^m$. Are there known ...

**4**

votes

**1**answer

267 views

### Fundamental class in K-theory and orientability

In ordinary homology, the classical results give the following situation:
for a compact, connected, topological manifold $M$ of dimension $n$ we have, for each ring $R$, that $H_n(M,M \setminus ...

**7**

votes

**1**answer

153 views

### Face poset of a subcomplex complement

Let $P$ denote the face poset of a simplicial complex, $\Delta$ the order complex of a poset, and $\sim$ homotopy equivalence. It's known that for any finite simplicial complex $\mathcal{K}$ that ...

**2**

votes

**0**answers

120 views

### 2-cells in the configuration space

My overarching question is why in the Artin presentation of the (geometrically defined) braid group there are no more than two relations. I've learned one way to prove this is using the fact that the ...

**3**

votes

**1**answer

217 views

### Motion planning algorithm

Consider a path connected topological space $X$, one can equip its path space $PX=\{ \gamma: [0,1] \longrightarrow X \; continuous\}$ with the compact open topology. We call a motion planning ...

**1**

vote

**0**answers

215 views

### Is this a “new” terminology in homology/cohomology theory?

I have the following question. For our research purpose, we have introduced the following concept:
Let $f:X\to Y$ be a continuous, disrecte and open mapping between two locally compact metric spaces. ...

**20**

votes

**4**answers

2k views

### Why the Dold-Thom theorem?

Dold-Thom Theorem: $$\pi_i(SP(X))\cong\tilde{H}_i(X)$$
It's pretty miraculous, no? I've seen its proof, where you show that the composition of the functors on the left-side satisfies the axioms of a ...

**4**

votes

**0**answers

167 views

### $n$-Fold Framed Functions

Suppose that $M$ is a manifold. One can consider a suitably constructed space of generalized framed Morse functions on $M$, let's call it $\mathrm{Fun}^\mathrm{fr}(M)$. This space is known to be ...

**3**

votes

**1**answer

263 views

### Classifying space for fibrations with Eilenberg-MacLane space as fibers

The following result seems to be frequently quoted:
Consider the fibration $K(\pi,n)=\Omega K(\pi,n+1)\to PK(\pi,n+1)\to K(\pi,n+1)$. Let $B$ be any topological space (which is not too pathologic). ...

**8**

votes

**2**answers

542 views

### Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?

The group $\mathbb{Z}_2$ acts on $S^7\times S^7$ by switching the coordinates with fixed point set $\Delta(S^7\times S^7)\cong S^7$. I want to know whether there are some other $\mathbb{Z}_2$ actions ...

**2**

votes

**2**answers

217 views

### A fibration of classifying spaces

Let $G$ be a Lie group, $N$ a closed connected normal subgroup. Let $BG$, $BN$, $B(G/N)$ be the classifying spaces of $G,N$ and $G/N$. Is there a fibration $BN\to BG\to B(G/N)$ ?
It seems that such a ...

**0**

votes

**0**answers

163 views

### Goodwillie tower of $\Omega^n$?

What are layers of the Goodwillie tower of the functor "n-th iterated loop space" from based spaces to based spaces? I know the answer for n=0.

**9**

votes

**1**answer

175 views

### Are Brown representable functors determined by restriction to finite complexes?

Assume two $CW$ complexes $X,Y$ give two functors $h_X=[-,X], h_Y=[-,Y]$ on the homotopy category of $CW$ complexes whose restrictions to the full subcategory of finite $CW$ complexes are naturally ...

**14**

votes

**2**answers

750 views

### Is homology finitely generated as an algebra?

If a differential graded algebra is finitely generated as an algebra, is its homology finitely generated as an algebra?
Is it easier if we impose any of the three conditions: characteristic zero; ...

**13**

votes

**1**answer

344 views

### Higher Cerf Theory

Morse functions on a manifold $M$ are defined as smooth maps $f:M \rightarrow \mathbb{R}$, such that at the critical points we can find local coordinates so that ...

**2**

votes

**0**answers

126 views

### The space of Fredholm operators as a classifying space

Is it true that the space of Fredholm operators on a separable Hilbert space is the classifying space for K-theory in the category of paracompact spaces?
Everyone quotes the theorem of Atiyah-Janich ...

**2**

votes

**0**answers

102 views

### When is the product of an infinite family of simplicial sets also a homotopy product?

The homotopy product of an infinite family of simplicial sets can be computed
by deriving the product functor sSetW→sSet, for example,
by performing the componentwise fibrant replacement using Kan's ...

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votes

**0**answers

117 views

### Fixed sets of orbit spaces

I've run across something that surprises me, so I'm wondering (1) Is it true? and (2) Is it well known? (And if the answers are affirmative, why didn't I know this already?)
Let $G$ be a compact Lie ...

**11**

votes

**3**answers

1k views

### (Very) High dimensional manifolds

Usually one regards manifolds up to dimension 4 as a part of low dimensional topology. There are plenty of various results which work only in low dimensional topology; especially in dimension 4. ...

**3**

votes

**1**answer

82 views

### cartesian product rigidity for the punctured open disc

Q1: Let $D^n$ ($n\geq 1$) be the n-dimensional open disk. If $D^n-\{0\}$ is homeomorphic to $X\times (0,1)$, for some topological space $X$, does it necessarily follow that $X$ is homeomorphic to ...

**1**

vote

**2**answers

185 views

### The cohomology groups of $\Omega U(n)$

Let $\Omega U(n)$ be the loop space of $U(n)$. Is it true that the cohomology groups $H^*(\Omega U(n); \mathbb{Z})$ are torsion-free? How can one calculate these groups?