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The spheres operad

I have a rather naive question. Consider the space of all maps $$ S^{j_1} \times S^{j_2} \times \cdots \times S^{j_k} \to S^n $$ for all possible natural numbers $n, k, j_1, \cdots , j_k$. This ...
Ryan Budney's user avatar
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11 votes
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203 views

Fundamental groups of reduced subgroup lattices

Let $G$ be a group. Its subgroup lattice, denoted $\Sigma G$, consists of all subgroups of $G$ partially ordered by inclusion. The topology of this poset is quite trivial, since it always has a ...
Vidit Nanda's user avatar
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11 votes
0 answers
1k views

Lyndon-Hochschild-Serre spectral sequence and cup products

First here is my setup: Let $W$ be some group, and $C$ a normal subgroup of finite index, and let $W/C=G$. Now let $L$ be a a $G$-module on which $C$ acts trivially, so in particular we get on action ...
Chris Birkbeck's user avatar
11 votes
0 answers
584 views

A proof of the gluing axiom of a TQFT

I posted the following question on math stackexchange but I have not received any answer. So I hope people here can help me. In the book Lectures on tensor categories and modular functors by Bakalov ...
user avatar
11 votes
0 answers
305 views

Generalized Classical Adjoints and Factorizations of the Characteristic Polynomial

This is idle noodling, and I'm prepared to learn that it's foolish as well as idle. But.... Let $M$ be an $n\times n$ matrix over, oh, let's say an algebraically closed field for now. There have ...
Steven Landsburg's user avatar
11 votes
0 answers
1k views

the first cohomotopy group of a P-adic solenoid

Greetings. I have a problem concerning the definition of the first cohomotopy group of P-adic solenoid. In [1], E. Spanier defines group operation on the set of homotopy classes of maps from X to n-...
Zvonimir's user avatar
  • 219
11 votes
0 answers
1k views

Original references for the homotopy groups $\pi_5(SU(3))$ and $\pi_4(SU(2))$?

For revision of a paper (http://arxiv.org/abs/1008.1189), I'd like to correct my references to the original work on aspects of the homotopy groups $\pi_5(SU(3))$ and $\pi_4(SU(2))$. I'm not a ...
Daniel Friedan's user avatar
11 votes
0 answers
561 views

How to get a Dehn-twist presentation of a periodic map of a Riemann surface?

Let $f:S\to S$ be a given periodic map of order $n$, and $(m_i,\lambda_i, \sigma_i)$ be the valency of the multiple point $p_i$ of $f$ ( $i=1,\cdots,r$ ). A classical result says such $f$ is ...
Jun Lu's user avatar
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10 votes
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420 views

What is the original source for the Goerss-Hopkins-Miller-Lurie theorem on tmf?

The central basic theorem of topological modular forms states that the structure sheaf of $\widehat{\mathcal{M}}_{ell}$ lifts to a sheaf of complex-oriented $E_{\infty}$-rings whose formal groups are ...
Doron Grossman-Naples's user avatar
10 votes
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213 views

Are topological theta series (taking values in tmf(N)) of lattices good for anything?

I'm going to start with Mike Hopkins' great survey article in the ICM on topological modular forms (https://arxiv.org/abs/math/0212397). In it, he outlines a construction, for even unimodular lattices,...
Mike's user avatar
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199 views

"Homotopy homomorphisms" of homeomorphisms of Euclidean space

For a topological group $G$, an older term for a map $BG \to BG$ is a "homotopy homomorphism". If $G$ is connected, taking based loops shows that a homotopy class of such a map is the same ...
skupers's user avatar
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10 votes
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219 views

Are multiples of representable homology classes still representable by smooth submanifolds?

Recently the following question comes up in my research. Suppose we have a closed compact connected smooth manifold $M,$ of dimension $d+c$ and an integral homology class $[N]$ induced by a compact ...
Zhenhua Liu's user avatar
10 votes
0 answers
308 views

Compact Lie groups are rational homotopy equivalent to a product of spheres

According to [1] and [2], it is “well-known” that a compact Lie group $G$ has the same rational homology, and according to [2] is even rational homotopy equivalent, to the product $\mathbb{S}^{2m_1+1} ...
Gro-Tsen's user avatar
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10 votes
0 answers
230 views

Lefschetz for topoi

The Lefschetz fixed-point theorem is a nice theorem in topology which counts the number of fixed points (counted with multiplicities) of a continuous map $f\colon X\to X$ using the traces of the ...
user1022117's user avatar
10 votes
0 answers
317 views

Near-ring spaces

$\newcommand{\la}[1]{\kern-1.5ex\leftarrow\phantom{}\kern-1.5ex}\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\...
Emily's user avatar
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10 votes
0 answers
187 views

A minimal $\mathbb Z/2$-invariant Morse function on $U(n)$

Consider the group $U(2n)$ of unitary matrices. This has two standard and important homomorphic involutions. The most famous one $A \mapsto \overline A$ is complex conjugation, with fixed-set $O(2n)$, ...
mme's user avatar
  • 9,580
10 votes
0 answers
455 views

Exotic analytic triangulations of $S^5$?

I would like to understand better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf http://www-math.mit.edu/~...
aglearner's user avatar
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10 votes
0 answers
325 views

Adams blue book lemma 17.14: computing a $\mathbb{F}_2$ basis for a filtration of $H\mathbb{Z}_*(bu \wedge bu)$

First off let me apologize for not being able to give all the context for this question. I'm learning how to do computations in stable homotopy theory and have been particularly spending a lot of time ...
Francis Baer's user avatar
10 votes
0 answers
446 views

Milnor's universal bundle as a colimit?

I have had Milnor's construction of the classifying space of a topological group explained to me on multiple occasions, and seen it described briefly in various places. But only now am I reading the ...
Nikhil Sahoo's user avatar
  • 1,225
10 votes
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342 views

Hodge structure and rational coefficients

Suppose $X$ is a complex projective variety with a model $X_\mathbb{Q}$ defined over the rational numbers. Then there is a rational de Rham lattice $H^k_{dR}(X_\mathbb{Q}, \mathbb{Q})\subset H^k(X, \...
Dmitry Vaintrob's user avatar
10 votes
0 answers
479 views

How do I produce a basis of cohomology?

Suppose I am discussing a smooth projective variety over an algebraically closed field with my friend on the phone and I want to make a statement about its $l$-adic cohomology (integral or rational). ...
user avatar
10 votes
0 answers
294 views

A certain semi-simplicial space

I would like to understand whether the following construction has been studied. Let me model the pointed sphere as $S^n = I^n / \partial I^n$, and let $\Omega^n (-) := map((I^n, \partial I^n), (-, *))$...
Oscar Randal-Williams's user avatar
10 votes
0 answers
176 views

Embedding 2-complexes null homotopically into 2-complexes

Whitehead's conjecture states that if $L$ is an aspherical 2-complex and $K$ is a subcomplex of $L$, then $K$ is also aspherical. It is known by work of Howie and Luft that if the Whitehead ...
user101010's user avatar
  • 5,349
10 votes
0 answers
214 views

For which $k$-types are $E_{n,m}$-algebras automatically $E_{n+1}$ algebras?

Recall that an $E_{n,m}$ algebra is an $A_m$ algebra in $E_n$ algebras. Here I index my $A_m$ algebras so that an $A_1$ algebra is pointed, an $A_2$ algebra has a unital multiplication, $A_3$ is ...
Noah Snyder's user avatar
  • 28.1k
10 votes
0 answers
217 views

Subvarieties with isomorphic complements

Let $X$ be a smooth irreducible projective variety over $\mathbb C$, $Y_1, Y_2$ are two closed smooth subvarieties. Assume $X-Y_1 \cong X-Y_2$, what do $Y_1$ and $Y_2$ have in common (at least ...
Zhiyu's user avatar
  • 6,622
10 votes
0 answers
399 views

Words and ranks

Let me state two problems that look very much alike. The first one can be solved putting together answers that different people have given to some questions I asked here a few weeks ago. The second ...
H A Helfgott's user avatar
  • 20.2k
10 votes
0 answers
393 views

Interpretation of determinants on commutative rings

In real Euclidian space, the result of the determinant can be interpreted as the oriented volume of the image of the unit cube under an invertible linear map. This interpretation conceptually depends ...
user's user avatar
  • 323
10 votes
0 answers
748 views

Models for Eilenberg-MacLane space K(Z,3)

Denote by $U(H)$ and $PU(H)$ the unitary and projective unitary groups on an infinite-dimensional Hilbert space $H$. Recall that $U(H)$ is contractible by Kuiper's theorem and that $PU(H)$ is a $K(Z,2)...
user46652's user avatar
  • 665
10 votes
0 answers
192 views

k-th Pontryagin class of $\Lambda^{2k}_{\pm}$ on an oriented $4k$-manifold

If $M^{4k}$ is an oriented Riemannian $4k$-manifold, then the star-operator splits the bundle $\Lambda^{2k}$ into $\pm 1$-eigenspace bundles denoted $\Lambda^{2k}_{\pm}$. I'm curious if anyone has ...
Brian Klatt's user avatar
10 votes
0 answers
281 views

How much is known about unstable homotopy of truncated projective spaces? - Reference request

I would like to know where about I can find the most updated results on the unstable groups $\pi_{2k+1}P_{k+1}$ and $\pi_{2k}P_k$. I think there would be definitely computations Mahowald's AMS memoir, ...
user51223's user avatar
  • 3,173
10 votes
0 answers
294 views

Simplicial model for $\Sigma^{\infty}_+ \Omega^{\infty}X$?

Let $Sp^{\Sigma}$ be the category of symmetric spectra, and let $Sp^{\Sigma}-alg$ be the associated category of augmented commutative algebras. These are simplicial categories. A question for experts:...
Nicholas Kuhn's user avatar
10 votes
0 answers
6k views

Atiyah's paper "Non-existent complex 6-sphere"

I'm trying to understand the main idea of Atiyah's proof (https://arxiv.org/abs/1610.09366). Although there were discussions on MO year ago I couldn't find answers to my questions. Consider the ...
Max Borovkov's user avatar
10 votes
0 answers
186 views

Countability assumption for good covers in Bott-Tu

In chapter II of their text Differential Forms in Algebraic Topology, Bott and Tu construct the Čech-De Rham complex with regards to an open covering indexed by some ordered and countable indexing set....
Amueller's user avatar
  • 253
10 votes
0 answers
414 views

Segal-Freed-Hopkins-Teleman = Atiyah-Hirzebruch/Leray-Serre?

Freed-Hopkins-Teleman (section 3.7) generalise Segal's (Proposition 5.3) spectral sequence for equivariant K-theory to more general local quotient groupoids (that is, topological groupoids locally ...
David Roberts's user avatar
  • 35.5k
10 votes
0 answers
435 views

The group analogue of the James construction

If $(X,e)$ is a based topological space, then the James construction on $(X,e)$ is the free topological monoid with unit $e$: $J(X)=\coprod_{n\geq 1}X^n /\sim $ where $(x_1,...x_{j-1},e,x_j,...,x_n)\...
Jeremy Brazas's user avatar
10 votes
0 answers
372 views

Steenrod Problem and realization of rational homology classes by manifolds

Steenrod's problem asks wheter a simplicial homology class of a topological space $x$, $$ x\in H_n(X, \mathbb{Z})$$ can be represented by a triangulation of an $n$-dimensional, ...
Nicolas Boerger's user avatar
10 votes
0 answers
268 views

Isomorphisms between minimal $A_\infty$-algebras having identical $k$-truncations

Let $A_m =(A,0,m_2,m_3,\dots)$ and $A_n=(A,0,n_2,n_3,\dots)$ be two $A_\infty$-structures on a vector space $A$. Assume that i) $A_m$ and $A_n$ are isomorphic, and ii) $A_m$ and $A_n$ have the same ...
Bashar Saleh's user avatar
10 votes
0 answers
468 views

Complex $K$-theory of extended powers of a Moore spectrum

Consider a Moore spectrum $S^n/p$. Has the $K$-theory of the extended powers of $S^n/p$ been computed? For some context: equivalently, I'd like to have the free $E_\infty$-algebra over $KU$ on the $...
Akhil Mathew's user avatar
  • 25.6k
10 votes
0 answers
246 views

$[K(\mathbb Z,4),\mathbb H\text{P}^{\infty}]$

The map $\mathbb H\text{P}^{\infty} \to K(\mathbb Z,4)$ representing a generator of $H^4(\mathbb H\text{P}^{\infty};\mathbb Z) = \mathbb Z$ is a rational equivalence. But is there any honest map in ...
Jens Reinhold's user avatar
10 votes
0 answers
387 views

When do contributions of $\pi_*(L_{K(n)}S^0)$ to $\pi_*(S^0)$ stabilize?

I apologize if this question is obvious as I am just beginning to learn this field. I am also abusing the dependance on $p$ for notational ease. The Chromatic Convergence theorem establishes that $S=\...
Pax's user avatar
  • 841
10 votes
0 answers
315 views

Is there a definition of an $\infty$-groupoid in HoTT whose terms are $n$-manifolds and whose higher morphisms are diffeomorphisms/isotopies/etc?

Suppose you want to work with TQFTs in homotopy type theory (HoTT). Working with $(\infty,n)$-categories, or even $(\infty,1)$-categories, is something that I gather is too difficult for HoTT at the ...
Andy Manion's user avatar
  • 1,474
10 votes
0 answers
276 views

6j symbols with Majorana indices

The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ symbols:...
Zitao Wang's user avatar
10 votes
0 answers
408 views

Combinatorial results by Poincaré duality

For the n-dimensional Torus, the k-th homology group (with integer coefficients) is isomorphic to the direct sum of $n \choose k$ copies of $\mathbb{Z}$. Poincaré duality thus gives us a somewhat ...
10 votes
0 answers
237 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 ...
Max's user avatar
  • 1,607
10 votes
0 answers
728 views

Homology of Lie groups

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...
user's user avatar
  • 293
10 votes
0 answers
813 views

On functoriality of the Leray spectral sequence

The Leray spectral sequence is functorial in the following sense: given a commutative square of spaces, $$\begin{matrix} A & \to & B \\ \downarrow & & \downarrow \\ C & \to & D ...
Dan Petersen's user avatar
  • 40.2k
10 votes
0 answers
365 views

diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...
Mikhail Katz's user avatar
  • 16.6k
10 votes
0 answers
193 views

Homotopy theory of suplattices

In Quillen's monograph Homotopical algebra where he introduced the notion of model category, he showed that if $C$ is a bicomplete category with enough regular-projectives in which either (*) every ...
Mike Shulman's user avatar
  • 66.8k
10 votes
0 answers
484 views

Space of embeddings of an $n$-ball into an $n$-manifold

Let $M$ be a smooth $n$-manifold without boundary, and let $B$ be the open unit ball in $\mathbb{R}^n$. I am trying to understand the space $\text{Emb}(B,M)$ of smooth embeddings of $B$ into $M$. ...
Lars's user avatar
  • 101
10 votes
0 answers
467 views

Galois action on $E_n$-operads

Let $E_n$ be the little $n$-cubes operad which acts on $n$-fold loop spaces (up to group completion, an $E_n$-action is precisely the data needed to perform an $n$-fold delooping). I am looking for ...
Akhil Mathew's user avatar
  • 25.6k

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