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

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8
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166 views

What is known about maps between loop spaces of Spheres? - Reference request

What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values ...
8
votes
1answer
292 views

Is the natural isomorphism $|FX_\bullet| \cong F|X_\bullet|$ lax symmetric monoidal?

Let $\mathcal{V_1}$ and $\mathcal{V_2}$ be cocomplete symmetric monoidal categories, each endowed with a cosimplicial object $\Delta^\bullet=\Delta^\bullet_{\mathcal{V}_i}:\Delta \to \mathcal{V}_i$. ...
7
votes
0answers
93 views

Homology of inverse limits over inverse systems more complicated than towers

Most textbooks discussions of homology of inverse limits of chain complexes consider only “towers,” i.e. inverse systems indexed by the natural numbers. I’d like to find a reference that explains what ...
1
vote
1answer
246 views

$G_1 \rtimes G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? $G_1 \times G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? [closed]

As summarized in the title, suppose there is an isomorphism between $G_1 \times G_2$ and $G_3$, is it always true that $BG_1 \times BG_2$ is homotopy equivalent to $BG_3$? If it is not always true ...
6
votes
0answers
56 views

Examples of maps with nontrivial Hopf invariant but Lusternik-Schnirelmann category of the cofiber doesn't increase?

Let $A$ be a suspension and $X$ be a space with Lusternik-Schnirelmann category $n$ and let $\alpha: A\to X$. It is easy to see that the cofiber $C_\alpha$ has $\mathrm{cat}(C_\alpha) \leq n+1$. One ...
5
votes
2answers
548 views

Can a Morse function define a unique structure on a closed manifold?

I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have ...
2
votes
0answers
86 views

Does attach-one-cell have a stable homotopy transfer?

Specifically, I am thinking of attaching one ordinary cell to an ordinary space in your favourite convenient category of spaces; so, given a cofiber sequence $$ \mathbb{S}^k \to_c X \to_p X', $$ on ...
10
votes
0answers
124 views

Existence of flat connections via characteristic classes, for nice groups

I have two questions about what I write below (which honestly seems pretty elementary). Is it true (more or less)? Is there a clean reference that I can cite. Let $G$ be a compact Lie group, $M$ a ...
4
votes
0answers
65 views

Criterion for a equalizer to be a homotopy equalizer in spaces

Let $f,g\colon X\rightarrow Y$ be maps between spaces. I am looking for criteria for the equalizer of $f$ and $g$ to be a homotopy equalizer and I am happy to get answers for whatever model category ...
8
votes
1answer
145 views

Cohomology operations on unoriented cobordism

In unoriented cobordism there exist stable cohomology operations looking similar to Steenrod squares (they were used by Quillen to compute the unoriented cobordism ring with its formal group law ...
6
votes
1answer
207 views

Can the Hochschild cochain complex be given the structure of a “homotopy BV algebra”?

In a 1993 letter, Deligne posed the following (paraphrased from a paper of Gerstenhaber and Voronov's): Conjecture (Deligne). The Hochschild cochain complex $CC^*(A)$ of an associative ...
2
votes
0answers
68 views

Can relative homotopy groups be represented as relative homology groups of some Moore complex?

Daniel Kan defined a combinatorial version of the homotopy group $\pi_n(X)$ of a simplicial set $X$ as the $(n-1)$st homology of the (non-commutative) Moore complex $\tilde{G}(X)$, where $G_iX$ is ...
6
votes
4answers
473 views

List of invariants that distinguish homotopy equivalent non-homeomorphic spaces

It is written on wikipedia article (https://en.wikipedia.org/wiki/Analytic_torsion) that the Reidemeister torsion is the first invariant that could distinguish between spaces which are homotopy ...
4
votes
0answers
166 views

Under what condition is a fiber bundle cobordant to the trivial bundle?

Let $E$ be the total space of a fiber bundle with base $B$ and fiber $F$, where $B$ and $F$ are smooth manifolds. Under what condition is $E$ unoriented cobordant to $B\times F$? And what happens ...
3
votes
0answers
163 views

How far can one reconstruct the boundary of a manifold M given its interior $int M$? [duplicate]

Suppose I keep in my pocket a manifold with boundary $M$ , and I provide you access to $int M := M \setminus \partial M$ up to homeomorphism/diffeomorphism. What can you deduce about $\partial M$? can ...
3
votes
0answers
65 views

Various definitions of the odd Chern character form

I am asking this question from my possibly defected memory, so the things below may not be accurate. I want to know how many different definitions of the odd Chern character form using differential ...
3
votes
0answers
149 views

Topology on $\mathcal{C}(X,Y)$ to work with homotopy

We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...
2
votes
2answers
213 views

For a universal covering morphism $p:E\rightarrow B$, how to prove $E$ connected implies $B$ connected?

Definition. An arrow $\alpha:A\rightarrow B$ in $\mathsf C=\mathsf{Fam}(\mathsf A)$ is said to be a covering morphism if there exists an effective descent morphism $p:E\rightarrow B$ that splits it, i....
5
votes
1answer
243 views

Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on ...
12
votes
1answer
136 views

The finiteness criterium $F$ under quasi-isometry

A group $G$ is defined to have $F$ if there exists a finite $K(G,1)$. This property is clearly not invariant under quasi-isometry as one can see from the trivial group and $\mathbb{Z}_2$. My question:...
1
vote
1answer
190 views

free group actions on a contractible topological space [closed]

Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $W$ be a contractible topological space with a free $\Sigma_k$-action (from the left). Let $X$ be a $CW$-complex and let $X^k$ be the ...
9
votes
1answer
298 views

When do non-exact functors induce morphisms on $K$-theory?

Let $\mathcal{A}$ and $\mathcal{B}$ be Waldhausen or exact categories, so that we can take the $K$-theory spectrum of $\mathcal{A}$ and $\mathcal{B}$. An exact functor $F: \mathcal{A} \to \mathcal{B}$ ...
2
votes
1answer
252 views

Collapse of Hirzebruch Spectral sequence

This question is actually about reading Adams' Stable Homotopy and Generalised Cohomology; in Part II chapter 2, there are two numbered lemmata (Lemma 2.5 contravariant, 2.14 covariant) to the effect ...
5
votes
2answers
235 views

When is $\mathbb C^d\setminus\mathcal Z$ simply connected?

Let $\Delta$ be a fan in the lattice $N\cong\mathbb Z^n$ with $d$ edges $\{\rho_1,\cdots,\rho_d\}$. Consider the co-ordinate ring $\mathbb C[x_1,\cdots,x_n]$. Let $\mathcal Z=\bigcup_C\mathcal V(x_i\ |...
6
votes
0answers
148 views

Identifying a certain element of $E^{2,0}_2$ with the orientation class of an oriented spectra

I'm interested in filling the details of the well known computation, via AHSS of $E^*(\mathbb{C}P^n)$; where $(E,x_E)$ is an oriented spectrum. It is supposed to be easy, yet every proof I've read ...
16
votes
0answers
505 views

Is the determinant of cohomology a TQFT?

If $M$ is an oriented $d$-manifold, let $D(M)$ denote the top exterior power of $H^*(M,\mathbf{C})$. Then $D(M_1 \amalg M_2) = D(M_1) \otimes D(M_2)$. Is there a good recipe for a map $D(M) \to D(N)$...
4
votes
1answer
387 views

“Polygons and gravitons” and Kodaira's theorem

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 470. At this point, he does some computations and obtains the conformal structure of the real ...
6
votes
1answer
309 views

Bar/Cobar Adjunction Between Modules and Comodules

There is a pretty well known, and widely written about, adjunction between augmented algebras and coaugmented coalgebras given by taking the bar construction on algebras and the cobar construction on ...
8
votes
1answer
273 views

When is bar-cobar duality an equivalence?

Let $A$ be an augmented differential graded algebra over a field $k$. I will write $BA$ for its bar construction (whose homology is $Tor^A(k, k)$). This is a co-augmented differential graded ...
5
votes
2answers
276 views

Serre fibration vs Hurewicz fibration

What are the simplest/ typical examples of a Serre fibration which is not a Hurewicz fibration? Is it something pathological? Sorry if the question is too elementary for MO.
2
votes
0answers
71 views

group actions of $S^3$ on the configuration space of projective plane

Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P^...
4
votes
0answers
147 views

Proving that an $E$-oriented manifold has an $E$-oriented normal bundle

This is the setting we are working in: $M$ is a closed, smooth $n$-manifold embedded in $\mathbb{R}^{n+k}$ with a chosen embedding $e\colon M^n\to \mathbb{R}^{n+k}$. It is $E$-oriented, for $E$ a ...
3
votes
0answers
94 views

h-principle on Hilbert manifolds

Gromov's h-principle is a powerful tool in studying various geometric structure on open, finite-dimensional manifolds. Is there any generalization of h-principle to (necessarily open) infinite-...
6
votes
0answers
148 views

Explicit diffeomorphism between an infinite dimensional sphere its product with itself

Let $S$ be an infinite dimensional sphere in a Hillbert space. As $S$ is homotopic to the product $S \times S$, then $S$ is diffeomorphic to $S \times S$ (for Hilbert manifolds, a homotopy ...
2
votes
1answer
251 views

induced group actions and covering maps on Eilenberg-Maclane space

Let $M$ be a finite $CW$-complex. Let $\Sigma_k$ be the symmetric group acting on $k$-letters. Suppose there is a free action of $\Sigma_k$ on $M$. Then we have a covering map $$ f:M\to M/\Sigma_k. ...
6
votes
0answers
97 views

mod $p$ homology module of unordered configuration spaces of the projective plane

Let $M$ be a manifold and $k$ be a positive integer. Let $F(M,k)$ be the $k$-th ordered configuration space over $M$, consisting of all ordered $k$-tuples of distinct points in $M$. Let $\Sigma_k$ be ...
3
votes
1answer
129 views

Framed bordism class of the circle has order $2$

As the question title says, how do I see that the framed bordism class of the circle has order two?
7
votes
0answers
111 views

Aityah-Patodi-Singer theorem in odd dimensions and Maslov triple indices

Let $W$ be a compact manifold with boundary and $D^W$ a graded Dirac type operator on $W$, of product type near the boundary acting on a vector bundle $E\to W$. One obtains a graded Fredholm operator $...
12
votes
1answer
295 views

Measuring the failure of pushforward to commute with Steenrod squares

Let $f \colon X \rightarrow Y$ be a map of topological spaces. Let's say that they are (closed) manifolds (not necessarily orientable), though to be honest I'm really interested in the more general ...
9
votes
2answers
354 views

Homotopy groups of Moore spaces

Is there anything known about the homotopy groups of the Moore spaces $M(\mathbb Z_m,n)$ if $m\neq 2$ and $n \geq 2$?
1
vote
1answer
238 views

Free Symmetric Operads and $\mathbb{S}$-modules

In the definition of operads, if we restrict our attention to $\mathbb{S}$-modules where the action by the symmetric groups is free, then the free operads have still an underlying free $\mathbb{S}$-...
8
votes
0answers
87 views

Are the unwound thin realization and fat realization homotopy equivalent?

This is a question about a theorem (proposition 2) in the article--On the homotopy type of classifying spaces Recall some definitions first: Given a category $\mathcal{C}$ internal in $\...
14
votes
1answer
217 views

Vector bundles with exactly one nonzero SW-class

I am interested in seeing examples of a space $X$ (preferably a closed smooth manifold, but any finite-dimensional CW-complex would also be of interest) with a vector bundle $\xi\colon E \to X$ on it, ...
2
votes
1answer
174 views

The compatibility of the Gysin sequence with mixed Hodge structures

Let $X$ be a compact complex $n$-manifold and $D$ be a smooth comdimension $1$ submanifold. Also let $U:= X\setminus D$ and $j$ be the inclusion of $U$ in $X$. Then it is well known that the ...
21
votes
2answers
954 views

The homotopy category is not complete nor cocomplete

I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts. What ...
2
votes
2answers
294 views

Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...
13
votes
3answers
544 views

Nice things that can be proved easily with characteristic classes

A bit of context for this question: as a project for my master's degree my supervisor asked me to understand the construction of Milnor's exotic spheres. After learning the heavy material (I knew very ...
2
votes
0answers
93 views

Mayer-Vietoris sequence for orbifolds

Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.
13
votes
2answers
497 views

Integral cohomology of $G/N(T)$

Let $G$ be a compact connected simple Lie group, $T$ a maximal torus, $N(T)$ the normalizer of $T$, and $W=N(T)/T$ the Weyl group. It is well-known that $H^*(G/T,\mathbb{Q})$ is the regular ...
7
votes
1answer
226 views

Thorough reference on regular homotopy

I would like to learn this topic of algebraic topology but I cannot find a relevant reference to answer my basic questions on the subject (for example, is there a Hurewicz theorem for regular homotopy ...