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

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2
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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
0answers
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
3answers
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
1answer
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
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1answer
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
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0answers
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
2answers
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
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1answer
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
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1answer
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
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0answers
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
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1answer
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
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0answers
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
1answer
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
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0answers
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
1answer
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
0answers
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
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0answers
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
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1answer
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
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0answers
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
1answer
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
2answers
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
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1answer
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
1answer
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
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1answer
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 ...
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0answers
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
1answer
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
2answers
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
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1answer
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
0answers
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 ...
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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
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2answers
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
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0answers
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
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0answers
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 ...
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vote
2answers
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 : ...
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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
1answer
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
1answer
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 ...
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1answer
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
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0answers
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
1answer
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
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2answers
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
4answers
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 ...