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

5
votes
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
3answers
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
0answers
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
0answers
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
1answer
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
0answers
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
3answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
2answers
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
3answers
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
2answers
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
1answer
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
0answers
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
2answers
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
0answers
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
0answers
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
3answers
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
1answer
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
2answers
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 ...