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

**2**

votes

**0**answers

164 views

### Cohomology spectral sequence over $k[t]$

I am trying to compute $H^*(X)$ for a (potentially large, finite, finitely filtered) simplicial complex $X$ using a cover $U_i$ of $X$.
I am building chain complexes for $X$ with a simplex that ...

**7**

votes

**0**answers

222 views

### A model category for E-infty algebras in a non-monoidal model category?

Given a suitable nice symmetric monoidal category $C$, symmetric monoidally enriched, tensored, and cotensored over a symmetric monoidal category $S$, and an operad $\mathcal{O}$ in $S$, we can ...

**2**

votes

**3**answers

364 views

### Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds

I would like to find a pedagogical reference where the classification, up to isomorphism, of principal $SU(2)$ bundles over a four-dimensional compact, oriented manifold is explained. In particular I ...

**2**

votes

**1**answer

185 views

### Showing left module actions are highly structured

For my own convenience I'll work in $\infty$-categories, feel free to answer in whatever framework best suits you. My question is essentially how to show, given an $E_\infty$-ring object $R$ in an ...

**0**

votes

**1**answer

133 views

### Ambient isotopy of the diagonal submanifold in product space

Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
...

**5**

votes

**0**answers

159 views

### homology theory for affine and projective algebraic sets?

Given $f_1,\ldots,f_r\in K[x_1,\ldots,x_m]$, resp. homogeneous $f_1,\ldots,f_r\in K[x_0,\ldots,x_m]$, is there a chain complex built from these polynomials, such that any polynomial map $\varphi\!: ...

**0**

votes

**2**answers

258 views

### integral or rational cohomology of real grassmannians

I have obtained that the cohomology rings
$$
H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k].
$$
Also
$$
H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...

**3**

votes

**1**answer

139 views

### homology of configuration spaces of non-compact manifolds

Let $M$ be a manifold.
Let $F(M,n)$ be the configuration space of $n$-tuples on $M$.
Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered ...

**9**

votes

**1**answer

331 views

### Reference request: Topology on the space of smooth compact submanifolds

In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ ...

**1**

vote

**0**answers

156 views

### Under which conditions the inclusion of a sub-simplicial set of the nerve of a category is a Joyal equivalence?

Let $i: X \to \mathrm{N}\mathcal C$ be a monomorphism in the category of simplicial sets, with $\mathcal C$ a category and $\mathrm{N}\mathcal C$ its nerve. I am looking for sufficient conditions (and ...

**7**

votes

**1**answer

150 views

### Citation: earliest incidence of the Borel localization theorem

The Borel localization theorem in (Borel) equivariant cohomology states that if $T$ is a torus and $M$ a smooth $T$-manifold, with fixed point set $M^T$, then upon localizing the coefficient ring ...

**6**

votes

**0**answers

252 views

### Twisted equivariant modular forms?

I'd like to know where I can find information about a class of objects which I think deserve to be called twisted equivariant modular forms. Let me guess a definition, indicate how it can be made more ...

**2**

votes

**1**answer

298 views

### pushforward of universal objects along canonical morphisms of stacks

The kind of question I'm interested in has the following flavor: having two moduli stacks with one being an enhanced (i.e more data) version of the other with the natural "forgetting" map between them ...

**1**

vote

**0**answers

96 views

### Dividing a n- cochain by a 1-cochain

Assume that $X$ is a path connected hausdorff topological space. Let $\alpha\in C^{1}(X,\mathbb{R})$ and $\beta\in C^{n}(X,\mathbb{R})$ be cochains in real singular cohomology. Asume that $\alpha ...

**4**

votes

**1**answer

198 views

### Bar Construction Model of Ring Spectrum Quotient

Suppose I am given a morphism $f:BG\to BGL_1(R)$ for $R$ some at least $E_1$-ring spectrum and $G$ a loop space. Then This corresponds, I believe, to an action of $G$ on $R$, coming from a morphism ...

**1**

vote

**0**answers

103 views

### Properties of “incomplete finite simplicial complexes”

Definition: We say that $K'$ is an incomplete finite simplicial complex if there exists a finite simplicial complex $K$ such that $|K'|=|K|\backslash Y$ where $Y$ is a union of some open faces of K.
...

**3**

votes

**0**answers

89 views

### Multiplicativity of combinatorial l classes

For closed smooth manifolds $M$ and $N$, the Hirzebruch $L$ class is multiplicative, i.e. $L(M\times N)=L(M)L(N)$. Is this property still true if $M$ and $N$ are assumed to be closed topological ...

**12**

votes

**1**answer

558 views

### Elliptic curves and connected components

Are there elliptic curves of positive rank with two real connected components
in which all the rational points lie only on one component?
Concrete examples are really appreciated.

**6**

votes

**0**answers

141 views

### Connection between quasifibrations and homotopy cartesian squares

Let me first fix the definitions.
A map $p\colon E\rightarrow B$ is called a quasi-fibration, iff the canonical inclusion $p^{-1}(b)\rightarrow hofib_b(p)$ is a weak equivalence for all for all $b\in ...

**11**

votes

**1**answer

410 views

### Is every degree 1 self-map a homotopy equivalence?

In a rather obscure article, I found (without proof) the following statement:
If $M$ is a closed orientable manifold, every degree $1$ map $f: M \rightarrow M$ is a homotopy equivalence.
Is this ...

**6**

votes

**2**answers

292 views

### Global Affine Flag Variety and Affine Flag Variety

There is a construction of a global affine flag variety over $\mathbb{A}^1$ (or another curve) $Fl_{\mathbb{A}_1}$ such that each fiber above $\epsilon \neq 0$ is isomorphic to a direct product of the ...

**1**

vote

**1**answer

319 views

### Isomorphism between a mapping class group and the fundamental group of a moduli space

For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...

**7**

votes

**1**answer

319 views

### Intuitive Aproach to Dolbeault Cohomology [closed]

(Duplicated from math.stackexchange)
I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. ...

**2**

votes

**1**answer

112 views

### Intersection of two real polynomial surfaces

Consider two real polynomials in three variables, defined on the 3-sphere, $S^3$. Is there some Bezout-type theorem, relating the intersection of two closed surfaces defined by these polynomials and ...

**0**

votes

**0**answers

97 views

### A naturality question concerning the universal coefficient spectral sequence

I am reading Hillman's book "algebraic invariants of links" and on page 20 he mentions the following universal coefficient spectral sequence.
Let X be a connected finite CW complex.Let $H$ be a ...

**1**

vote

**1**answer

294 views

### Does the singular cohomology for a metric space of finite topological dimension vanish in high dimensions?

It is known that by applying the universal coefficient theorem, the singular cohomology of closed manifold with coefficient $\mathbb{Z}_2$ vanishes in high dimensions. But for a metric space $M$ with ...

**6**

votes

**2**answers

276 views

### Is every $S^3$ block bundle over $S^4$ a fiber bundle?

I am interested in the difference between block bundle and fiber bundle.
Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.
A block diffeomorphism of $\Delta^p\times M$ is a ...

**2**

votes

**1**answer

173 views

### Complexification of real k-theory gives index $2$ subgroup of complex k-theory

We have $\widetilde{KO}(S^4) \cong \mathbb{Z}$ and $\widetilde{K}(S^4) \cong \mathbb{Z}$. There is a map $i:\widetilde{KO}(S^4) \rightarrow \widetilde{K}(S^4)$ that takes a stable vector bundle to ...

**5**

votes

**0**answers

143 views

### G-spaces and SG-module spectra

This question is related to the one here, but has a slightly different angle.
Let $G$ be a topological group and let $X$ be a $G$-space. Taking the suspension spectrum $\Sigma^{\infty}_+ X$ (in my ...

**0**

votes

**0**answers

69 views

### cohomology of labelled configuration space & relation with braid space

Let $M$ be a manifold and $(X,*)$ be a pointed topological space. ( If we want, we can let $M=S^2,S^1\times \mathbb{R},etc.$)
Let $F(M,k)$ be the ordered configuration space of $k$-tuples on $M$.
...

**4**

votes

**2**answers

244 views

### Reference for homotopy (co)limits of (co)chain complexes via totalization of double complexes

It seems to be a well-known fact that homotopy (co)limits
of (co)simplicial diagrams of nonnegatively graded
(co)chain complexes in (Grothendieck) abelian categories
can be computed by using the ...

**3**

votes

**2**answers

219 views

### cohomology algebra of braid spaces, configuration spaces

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, Chapter 5, 6, 7, 8, 9, 10, 11, the cohomology algebra $H^*(B(\mathbb{R}^{n+1},p),\mathbb{Z}_p)$, for ...

**0**

votes

**1**answer

122 views

### cohomology version of Cartan-Leray spectral sequence that deduces cup product

On page 338, A User's Guide to Spectral Sequences. 2nd Edition, by John McCleary, Theorem 8.9, there is a Cartan-Leray spectral sequence for homology:
If $X$ is a connected pace on which the group ...

**3**

votes

**1**answer

138 views

### Mod 2 Totaro spectral sequence for non-orientable manifolds

I've been reading Burt Totaro's article "Configuration spaces of algebraic varieties", and I have a question regarding the main theorem (Theorem 1 in page 3 of the pdf). The theorem in question gives ...

**5**

votes

**1**answer

149 views

### Reference for Mod 2 cohomology of $BZ_{2r}$ in terms of Stiefel-Whitney Classes

I was hoping for an explicit reference to the description of the mod 2 cohomology of a cyclic group $C_{2r}=\langle t \rangle$ of even order in terms of Stiefel-Whitney classes, i.e., that
...

**13**

votes

**1**answer

239 views

### Embedding of two-dimensional CW complexes which induces a zero homomorphism on second homotopy groups

I am interested in the following question which I asked at MSE and did not get any clues. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective ...

**1**

vote

**1**answer

147 views

### fiber sequence of principal bundles

Let $G$ be a group, either a Lie group or a discrete group. Let a principal $G$-bundle
$$
G\to E\to B,$$
then $B=E/G$, the orbit space under action of $G$.
Let $BG$ be the classifying space of $G$.
...

**12**

votes

**0**answers

351 views

### Smooth 4-manifolds with $E_8$ intersection form

Does there exist a closed orientable smooth 4-manifold $M$ whose intersection form is the $E_8$-form? Here by the intersection form I mean the $\mathbb{Z}$-valued bilinear form on ...

**7**

votes

**1**answer

352 views

### Generalized Thom spectra

I am currently trying to wrap my head around all kinds of different definitions of the notion of (generalized) Thom spectrum. My setup is as follows: Suppose I have a commutative (symmetric) ring ...

**4**

votes

**0**answers

265 views

### Very frustrated reading a proof of the faithfulness of Artin's representation of braid groups

I am reading BRAID GROUPS, FREE GROUPS, AND THE LOOP SPACE OF THE 2-SPHERE by F.R. Cohen and J. Wu and here is an extract of the paper:
(The proof is not finished yet but I am very confused by now.)
...

**10**

votes

**0**answers

157 views

### Stable range of some classifying spaces and iterated loop spaces

Galatius (in his talk) has made very interesting remarks about the stable range of some classifying spaces of groups. To be more concrete, I will mention two examples to illustrate his (?) point of ...

**1**

vote

**2**answers

101 views

### Reference for the proof of a neighbourhood characterisation of cofibrations

I am interested in a reference for the proof of the following
theorem for $A,X$ being CGWH topological spaces.
Let $A\subset X$ be a closed subspace,
such that there exists a continuous $\phi : ...

**2**

votes

**0**answers

138 views

### Twisting of the power functor

Let $k$ be a field of characteristic $p$ and $D^b(k)$ be the infinity (equivalently, DG) category of perfect complexes over $k$. Let $C_p(=\mathbb{Z}/p)$ be the cyclic group on $p$ elements. For a ...

**0**

votes

**1**answer

147 views

### Homology of a finite disjoint union of open cells

Let $X$ be a topological space. Assume that $X$ admits a finite decomposition of the form $X=\bigsqcup\limits_{i=1}^n V_i$ where each $V_i$ is homeomorphic (in the subspace topology of $X$) to an open ...

**4**

votes

**1**answer

240 views

### Relation between cohomology of ordered and unordered configuration spaces

Let $M$ be a manifold. Then $F(M,k)/\Sigma_k$, the unordered configuration space of $k$ points, is obtained as a quotient of $F(M,k)$, the ordered configuration space of $k$ points, by the group ...

**-1**

votes

**1**answer

91 views

### cohomology algebra of unordered configuration space on Euclidean space

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, page 210 (the preface part before contents):
Line 2: ... is used to compute the precise algebra ...

**4**

votes

**0**answers

133 views

### cohomology algebra of unordered configuration space with coefficients the finite fields

in the paper The cohomology algebra of unordered configuration spaces (Y. Félix, D. Tanré, J. London Math. Soc., 2005), Theorem 4:
Let $M$ be an odd-dimensional, compact, closed, oriented manifold. ...

**5**

votes

**1**answer

313 views

### The function algebra $C^{\infty}(M\#N)$ of the connected sum of two spaces

Operations such as taking union or Cartesian products of spaces have direct analogues in term of algebra of functions on them (direct sum and tensor product, respectively),
my question is:
Is there ...

**28**

votes

**2**answers

973 views

### Is there a “simplification” functor in algebraic topology?

Recall that a space (=CW complex) is called simple if it is connected, the fundamental group is abelian, and the fundamental group acts trivially on all higher homotopy groups. Call Simp(X) a ...

**10**

votes

**4**answers

891 views

### The periodic values in Bott periodicity

After Bott periodicity is proved, one still has to compute the stable values. For the unitary group $U$, this is easy since you can get away with just $\pi_0$ and $\pi_1$. However, I'm having ...