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

**5**

votes

**1**answer

92 views

### Monoidal structures on modules over derived coalgebras

Given a Hopf-algebra $H$ (over a commutative ring), it is a classical fact that its category of (left) modules is monoidal, even if $H$ is not commutative. Given two left modules $M$ and $N$, we can ...

**2**

votes

**1**answer

82 views

### Two model structures of some category

Let $\mathscr{C}$ be a category and $(W,C,F)=$(weak equivalence,cofibration,fibration) is some model structure of $\mathscr{C}$. Another model structure of $\mathscr{C}$, $(W_{S},C,F_{S})$ are called ...

**5**

votes

**0**answers

138 views

### What does it mean to say the first Goodwillie derivative of $TC$ is $THH$?

A paradox:
Goodwillie calculus considers only finitary functors.
$TC$ isn't finitary.
Yet in some sense $\partial(TC) = \partial(K) = THH$ is the crux of the Dundas-Goodwillie-McCarthy theorem.
(...

**11**

votes

**1**answer

303 views

### Homotopy orbits, spectra and infinite loop spaces

Let $X$ be an (naive) $O(n)$-spectrum (I'm choosing to work with orthogonal spectra). I've recently come across the following results,
$$(S^{n-1} \wedge X)_{hO(n)} \simeq X_{hO(n-1)}$$
and
$$\Omega^...

**14**

votes

**0**answers

167 views

### How many cells are needed in a simplicial structure of $\mathbb{S}^n$ to induce all of $\pi_n(\mathbb{S}^m)$

Serre proved, that for (allmost) all $n,m\in\mathbb{N}$ the homotopy groups $\pi_n(\mathbb{S}^m)$ are finite, so - using simplicial approximation - for $n, m$ fixed there is a finite cell ...

**2**

votes

**1**answer

122 views

### Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures

Let $S=S_{g,b}$ be a compact orientable surface with genus $g$ and $b$ boundary components, such that $\chi(S)=2-2g-b<0$. Let $Q=\{x_1,\ldots , x_n\}$ be a set of $n$ distinguished points in the ...

**7**

votes

**1**answer

141 views

### Simplicial nerve functor commutes with opposites

There are two "opposite" functors:
$$ op_\Delta\colon sSet\to sSet$$
and
$$op_s\colon sCat\to sCat.$$
The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite ...

**8**

votes

**2**answers

275 views

### Torus action implying infinite fundamental group

Suppose that a $d$-dimensional torus $T$ acts smoothly and effectively on an $n$-dimensional closed manifold $M$. What conditions on $d$ and $n$ imply that $\pi_1(M)$ must be infinite?
Consider the ...

**3**

votes

**1**answer

251 views

### Thom space, homotopy group and cohomology group

In Thom's 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés ...

**2**

votes

**0**answers

58 views

### Relation of nerve of groupoid and 1st Postnikov object

Let $B$ be a fibrant simplicial set and let $B^{(1)}$ be its 1st Postnikov object. Let $\mathscr{G}$ denote a groupoid such that $\mathrm{Obj}(\mathscr{G})=B_{0}=B^{(1)}_{0}$ and $\mathrm{Aut}_{\...

**9**

votes

**3**answers

330 views

### Induced maps on homotopy groups by self maps of $\mathbb{CP}^n$

Let $f:\mathbb{S}^2\to \mathbb{S}^2$ with degree $d$.
It is well known that the induced map $$f_\ast:\pi_3(\mathbb{S}^2)=\mathbb{Z}\to \pi_3(\mathbb{S}^2)=\mathbb{Z}$$ is given by multiplication by $...

**1**

vote

**0**answers

109 views

### Representing curves using words

I am trying to understand how in this paper https://arxiv.org/abs/1412.0101 he represents curves with words. This is on page 10 of the paper.
Assume that two piecewise smooth closed curves $\gamma_1$ ...

**4**

votes

**0**answers

125 views

### Deforming a section to a section without zeros

Let $M$ be an oriented manifold of dimension $n$. Suppose furthermore that $E$ is an oriented vector bundle of rank $n-1$ over $M$. Let $s$ be a section of $E$ transversal to the zero section in $E$. ...

**9**

votes

**1**answer

208 views

### Fourth obstruction, Pontryagin and Euler class

Assume the first three obstruction classes of a rank 4 vector bundle vanish and look at the fourth obstruction class. This fourth obstruction class can be decomposed as the Euler class and the first ...

**11**

votes

**0**answers

130 views

### A geometric interpretation of the odd-primary Kervaire elements

Let $\Omega^\mathrm{fr}_\ast \cong \pi_\ast S$ denote the graded ring of cobordism classes of framed manifolds, which, by the Pontryagin-Thom construction, is isomorphic (as a graded ring) to the ...

**6**

votes

**3**answers

206 views

### $S$-dual of filtered spectra

I hope this is research level. Suppose $E$ is the direct limit of finite spectra, say $E=\mathrm{colim }\ E_i$, which itself is not finite. I wonder how much and under which conditions the inverse ...

**8**

votes

**1**answer

93 views

### A relative Kuiper theorem

Let $(H_0, \langle \,,\,\rangle_0)$ be a real separable Hilbert space,
and let $(H_1, \langle \,,\,\rangle_1)$ be a Hilbert space such that $H_1 \subset H_0$ is dense and such
that the inclusion $(...

**9**

votes

**0**answers

139 views

### Generalize $\mathbb Z/p$-space for irrational $\alpha$

A free $\mathbb Z/p$-space is a topological space $X$ with an action $\varphi$ such that $\forall x\in X$ $\varphi^p(x)=x$ but $\varphi(x)\ne x$.
I would like to generalize this notion from $\frac 1p$ ...

**5**

votes

**1**answer

187 views

### Algebraic models of non-simply connected spaces in string topology

I want to find some algebraic models relating string topology to Hochschild and cyclic homologies. If the space $X$ is simply-connected and we are working over rational numbers, we can use Sullivan ...

**7**

votes

**0**answers

165 views

### What is $SL(2,\mathbb{R})$-Chern-SImons Theory?

I found in physics that Chern-Simons theory is closely related with three dimensional gravity.
From this paper Three Dimensional Gravity Revisited, the author talks about the Chern-Simons for
$$\...

**0**

votes

**0**answers

24 views

### critical points relevant to the lowest order non-perturbative correction

I am interested in the Hyperasymptotics of multidimensional integrals of the form
$$\mathcal{I}(\lambda) = \int_{\mathbb{R}^n} dz_1 \wedge dz_2 \wedge \dotsi \wedge dz_n \, g(z_1,\dotsi,z_n) \, e^{\...

**-2**

votes

**1**answer

382 views

### Generalizing the Madsen-Weiss Theorem via the scanning map $\mathscr{C}(M,\mathbb{R}^{\infty})\to\Omega^{\infty}AG^+_{\infty,d}$

The Madsen-Weiss Theorem, as described by Hatcher, states that there is an isomorphism $H_*( \mathscr{C}_{\infty})\cong H_*(\Omega_0^{\infty}AG^+_{\infty,2})$ where $\Omega_0^{\infty}AG^+_{\infty,2}$ ...

**3**

votes

**1**answer

146 views

### Formal complex manifold without dd^c

Is there an example of compact complex manifold, which is formal, but does not admit complex structure satisfying $dd^c$-lemma?

**4**

votes

**2**answers

342 views

### Find a surface or 3-manifold whose fundamental group is $(\mathbb{Z}/n\mathbb{Z}) \rtimes (\mathbb{Z}/2\mathbb{Z})$

I know by Van Kampen's Theorem that we can obtain $\pi_1(S_1 \vee S_1) = \mathbb{Z} * \mathbb{Z}$, so I am wondering if we can construct a surface or 3-manifold whose fundamental group is $\mathbb{Z}...

**8**

votes

**1**answer

190 views

### Reference for: $p$-primary component of $\pi^S_k$ is $\Bbb Z_p$ when $k=2l(p-1)-1$

I remember coming across this result some time ago but I am having trouble finding a reference for it. It goes something like this:
Let $p$ be a(n odd?) prime, then the $p$-primary component of $\...

**5**

votes

**1**answer

277 views

### Kunneth formula for semidirect product

I wonder if the following Kunneth formula for semidirect product is valid
$$ H^n(N\rtimes_\phi G;\mathbb{Z}) = \sum_{i+j=n} H^i(G; H^j(N;\mathbb{Z})),$$
where $H^*$ is the group cohomology and $G$ has ...

**2**

votes

**0**answers

52 views

### Extension of a given section and obstruction cocyles

Let $p:E \to X$ be a fibration with the fiber $F$ where $X$ is a CW-complex. Denote by $U$ the set $U:=D^n \times \{0\} \cup S^{n-1} \times I$ (part of a cylinder) and let $\tilde{f}:U \to E$ be a ...

**0**

votes

**0**answers

79 views

### Lifting action to the sphere

Consider the free action of $\mathbb{Z}_2$ on $\mathbb{C}P^n$( $n$ is odd ) by $$[z_0,\dots, z_n]\rightarrow [-\overline z_1,\overline z_0,\dots,-\overline z_{n-1},\overline z_n].$$ Consider the orbit ...

**8**

votes

**1**answer

117 views

### Does the Serre spectral sequence of the Fadell-Neuwirth fibration collapse if there is a cross-section?

I had asked a vague question in MSE where a useful pointer to the Leray-Hirsch theorem was mentioned by Mike Miller in the comments, but received no answers. Here I will specialize to an interesting ...

**2**

votes

**1**answer

65 views

### On the entries of a matrix representation for a boundary operator of a persistence module

In equation 6 of Computing Persistent Homology (page 8), the authors put forward the following identity:
$$\deg \hat{e_i}+\deg M_k (i,j)=\deg e_j$$
Where $\hat{e_i}$ and $e_j$ are elements of ...

**6**

votes

**2**answers

259 views

### Homology spectral sequence for function space

The question is in the title. Suppose that $X$ and $Y$ are two pointed connected CW-complexes. I was wondering if there exists a spectral sequence computing the homology of the function space $$H_{\...

**0**

votes

**0**answers

108 views

### How does one obtain a classification of reducible holonomy groups from Berger's classification?

How does one obtain a classification of reducible holonomy groups from Berger's classification with de Rham's decomposition theorem? It was suggested to me that I should requiring that each factor in ...

**5**

votes

**1**answer

319 views

### Obstructions for the lifting problem after a pull-back

This is a cross-post from a MSE question which received no answers. Beware that the notation here is a little different.
Consider the following lifting problem(s):
$\require{AMScd}$
\begin{CD}
&...

**4**

votes

**0**answers

96 views

### Does real formality descend to rational formality for operads?

A classical theorem in rational homotopy theory says that a space is formal over $\mathbb{Q}$ iff it is formal over any field of characteristic zero. In other words, the algebra $A^*_{PL}(X)$ is ...

**1**

vote

**0**answers

41 views

### Weighted cancellation norm of a word computation

A symmetric set without identity $S$ is a set with a bijective function $inv : S \rightarrow S$ with no fixed points such that $inv(inv(x)) = x$ for any $x \in S$.
We say that two disjoint pairs $\{...

**6**

votes

**1**answer

199 views

### Is there a relationship between norms/transfers in equivariant homotopy theory and norms in the Tate construction / ambidexterity?

In homotopy theory, the word "norm" is commonly used in two different ways (well, surely there are other ways, but these two have a particular familial resemblence).
Let $G$ be a finite group. A $G$-...

**3**

votes

**0**answers

107 views

### Is the simplicial objects functor a comonad?

Let $T$ be the functor of simplical objects $[\Delta^{\mathrm {op}},-]:\mathrm{Cat} \to \mathrm{Cat}$. I am trying to construct counit and comultiplication maps $\eta$ and $\mu$ to make $(T,\eta,\mu)$ ...

**7**

votes

**1**answer

120 views

### When does a map of spaces deloop a closed subgroup inclusion?

I believe Kan showed that any connected CW complex is the delooping of a topological group. I'm interested in the relative question:
Question: Let $Y \to X$ be a map of connected CW complexes. Under ...

**2**

votes

**1**answer

563 views

### Show that the rational cohomology ring $H^*(M;\mathbb{Q})$ needs at least two generators

Let $M$ be a simply connected closed Riemannian manifold. How does one find a necessary condition going both ways that may be imposed on $M$ (perhaps on the curvature of $M$ and on torsion) which ...

**19**

votes

**2**answers

537 views

### Can one compute the fundamental group of a complex variety? Other topological invariants? [duplicate]

Given a system of polynomial equations with rational coefficients, is there an algorithm to compute the geometric fundamental group of the variety defined by these equations? I'm interested in both ...

**10**

votes

**3**answers

245 views

### Unbiased Hopf algebras

In category theory, a notion of monoidal category in which every sequence $X_1, \ldots , X_n$ ($n\ge 0$) of objects has a specified product is called an ``unbiased monoidal category'' (see Section 3.1 ...

**3**

votes

**2**answers

154 views

### Best notation for fibrant/cofibrant replacement

In Quillen's original text on model categories (homotopical algebra) he uses $Q$ and $R$ to denote cofibrant and fibrant replacement respectively.
This notation has been used by several other ...

**6**

votes

**1**answer

205 views

### to compare cohomologies of fibers of two fiber bundles

Consider the following commutative diagram of the fiber bundles $%
F\rightarrow E\rightarrow B$ and $F^{\prime }\rightarrow E^{\prime
}\rightarrow B^{\prime }$ where $B^{\prime }$ is simply connected ...

**10**

votes

**0**answers

244 views

### Is the $\hat{A}$-genus invariant under crepant birational maps between smooth algebraic varieties?

The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ ...

**16**

votes

**2**answers

414 views

### What is the weakest negative curvature condition ensuring a manifold is a $K(G,1)$?

The only statement I'm sure of is that any hyperbolic or Euclidean manifold is a $K(G,1)$ (i.e. its higher homotopy groups vanish), since its universal cover must be $\mathbb H^n$ or $\mathbb E^n$. ...

**3**

votes

**0**answers

47 views

### cohomology of the orbit space of a compact totally disconnected group action on a paracompact space

It is well-known the next theorem at Chapter III, Theorem 7.2. in Bredon's Introduction to compact transformation groups book.
Theorem: Let $X$ be a paracompact $G$-space with $G$ finite and let $\pi:...

**4**

votes

**0**answers

270 views

### Nonabelian Poincare duality

Nonabelian Poincare duality is introduced by Jacob Lurie in "Higher Algebra", Section 5.5.6.
It seems, that my question is closely related to this definition.
Question: what can one say about the ...

**11**

votes

**3**answers

573 views

### Existence of non-null-homotopic map from $M^n$ to $S^{n-1}$

Let $M^n$ be compact, connected, oriented $n$-dimensional smooth manifold without boundary, the Hopf degree theorem states that the homotopy class of continuous maps from $M^n$ to $S^n$ is classified ...

**26**

votes

**1**answer

414 views

### What is the minimal dimension of a complex realising a group representation?

This question is inspired by this one, which was about representations that can be realised homologically by an action on a graph (i.e., a 1-dimensional complex).
Many interesting integral ...

**6**

votes

**2**answers

378 views

### Is $\Bbb S^2 \times \Bbb S^4$ symplectic?

I'm playing around with products $M = \Bbb S^{n_1} \times \Bbb S^{n_2}$, and a quick computation using the Künneth formula tells us that if $(n_1,n_2)$ is not $(1,1)$ or $(2,4)$, $M$ is not symplectic ...