Questions tagged [at.algebraic-topology]
Homotopy theory, homological algebra, algebraic treatments of manifolds.
8,228
questions
1
vote
0
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143
views
Example of open manifold with no free integer homology non-homeomorphic to a ball
I would like to state that if an open oriented even-dimensional (complex) manifold $M$ is such that $dim(H_k(M,\mathbb{Z}))=0$ for $k>0$, and 1 for $k=0$, then $M$ is homeomorphic to an open ball.
...
35
votes
3
answers
1k
views
Second Betti number of lattices in $\mathrm{SL}_3(\mathbf{R})$
We fix $G=\mathrm{SL}_3(\mathbf{R})$.
Let $\Gamma$ be a torsion-free cocompact lattice in $G$. Is $b_2(\Gamma)=0$?
Here the second Betti number $b_2(\Gamma)$ is both the dimension of the ...
2
votes
0
answers
71
views
Topological Shape Operator More Sensitive than Inverse Limits
This is a very general sort of question, and the use of the phrase 'shape operator' is a bit sloppy since there is already an established "shape theory." But what I have are topological spaces that ...
5
votes
1
answer
207
views
Cellularity of anodyne extensions?
Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts)
If not, is there a known counterexample?
Similarly, does ...
6
votes
1
answer
189
views
The Stone-Čech compactification of the fixed point set
Let $G$ be a discrete group and $X$ be a Tychonoff $G$-space. Then there
exists a $G$-action on Stone-Čech compactification $\beta X$. If the
fixed point set $X^{G}\neq \emptyset $, then the Stone-...
4
votes
0
answers
279
views
Generalized Postnikov square
Following Wikipedia (https://en.wikipedia.org/wiki/Postnikov_square), a Postnikov square is a certain cohomology operation from a first cohomology group $H^1$ to a third cohomology group $H^3$, ...
9
votes
2
answers
582
views
Künneth formulas/theorem for bordism groups and cobordisms?
We are familiar with Künneth theorem:
The Kunneth formula is given by $R$ as a ring, $M,M'$ as the R-modules, $X,X'$ are some chain complex. The Kunneth formula shows the cohomology of a chain ...
5
votes
1
answer
189
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Relating bordism groups of different dimensions
Let
$M_d$
be a $d$-manifold generator of a subgroup of bordism group
$$
\Omega_d^{G},
$$
or further generalization
$$
\Omega_d^{G}(K(\mathcal{G},n+1)),
$$
which $G$ is the given structure ...
3
votes
1
answer
138
views
Bordism invariants vanishes in a lifted twisted $Pin^- \times Spin$-structure
It looks to me that the bordism group
$$\Omega_3^{SO} (B(O(2) \times SO(3))) \tag{1}$$
(whose Pontryagin dual for the manifold generator) contains at least a nontrivial invariant:
$$
w_1(O(2))\big(...
4
votes
1
answer
349
views
(Co)bordism invariant of Eilenberg–MacLane space becomes vanished
Consider a (co)bordism invariant
$$
u_2 Sq^1 u_2+Sq^2 Sq^1 u_2
$$
obtained from
$$
\Omega^5_{O}(K(\mathbb{Z}/2,2)).
$$
Here $u \in H^2(K(\mathbb{Z}/2,2),\mathbb{Z}_2)$. The $K(\mathbb{Z}/2,2)$ is ...
4
votes
1
answer
793
views
homologies of some subsets of ${R}^{n}$
This might be something well-known.
For $1\le k\le n$, let $A(n,k)\subset\mathbb{R}^{n}$ be the set of points $x=(x_{1},...,x_{n}%
)$ with at least $k$ distinct coordinates. Then what are the ...
4
votes
1
answer
381
views
null-bordant vs null-homologous sub-manifolds of $\infty$-d spaces/CW complexes
$\require{AMScd}$
Preliminaries: Let $\Sigma$ be a closed manifold, $X$ be a CW complex and $f:\Sigma \to X$ be a map. We say that the pair $(\Sigma,f)$ is null-homologous (over $\mathbb{Z}_2$) if $...
1
vote
1
answer
385
views
Formality of surfaces
The de Rham dg algebra $\Omega(F)$
of a closed orientable surface $F$
is formal
(that is, weakly equivalent to its cohomology algebra).
This is a special case of the fact of formality of Kähler ...
2
votes
0
answers
67
views
Order relation between cohomology groups
We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex
$$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...
6
votes
1
answer
290
views
HKR generalized character theory question regarding a certain construction
In that paper https://web.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf Hopkins-Kuhn-Ravenel introduced the idea of generalized character corresponding to a complex-oriented cohomology ...
4
votes
0
answers
165
views
Complex projective algebraic variety, moduli space of flat connections, and instantons
In Looijenga's work below, if I understand correctly, it shows that
Statement 1: At an algebraic variety, the moduli space of SU($N$) flat
connections on a 2-torus $T^2$ is given by the space of ...
5
votes
0
answers
240
views
Homotopy functor calculus vs functor calculus in additive categories
Consider the Goodwillie calculus of a homotopy functor $F : \mathrm{Sp} \to \mathrm{Sp}$, where $\mathrm{Sp}$ denotes an appropriate model for spectra (... orthogonal spectra for instance).
Then ...
5
votes
0
answers
67
views
Does the theorem that genera vanishing on even-dim complex projective bundles are elliptic also apply for integral-valued genera?
Ochanine proved in this paper that for genera taking values in $\mathbb{Q}$-algebras, vanishing on even-dimensional projective bundles is equivalent to being an elliptic genus (i.e. a specialization ...
5
votes
2
answers
261
views
Naturality of PD model of a CDGA
In the paper "Poincaré duality and commutative differential graded algebras", Lambrechts and Stanley constructed PD model for cdga with simply connected cohomology. My question is: if $A$ and $B$ are ...
5
votes
0
answers
268
views
Betti sum, cup-length, Lusternik-Schnirelmann category, and critical points
I am trying to make some order in the notions of cup-length, sum of Betti numbers, LS category, critical points of functions.
Let $M$ be smooth closed compact manifold. We denote by Crit($M$) the ...
1
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0
answers
410
views
On Lefschetz theorem and sum of Betti numbers as lower bounds for fixed points
Let $M$ be a closed manifold with holomorphic cell decomposition (if it is complex), or at least with only even cohomology. In particular, its Euler characteristic is equal to the sum of its Betti ...
3
votes
0
answers
90
views
Cohen's definition of loop operations
I have a problem with understanding Cohen's definition, as written in "The Homology of Iterated Loop Spaces" by Cohen, Lada and May (Part III, chapter 5, at the end of chapter).
So in order to define ...
23
votes
1
answer
445
views
To what extent can we characterise the image of the topological Chern character?
For a finite CW complex $X$, the Chern character gives an isomorphism
of finite-dimensional vector spaces:
$$
ch : K^*(X)\otimes \mathbb{Q} \to H^*(X, \mathbb{Q}).
$$
The vector space $V = H^*(X, \...
7
votes
0
answers
224
views
The limitation of $G$ and loop group $\Omega G$ in Atiyah's and Donaldson's work on Instantons
In Atiyah's work [Ref. 1], Atiyah states that "Essentially we shall show (at least for $G$ a classical
group and probably for all $G$) that Yang-Mills instantons in 4D can be naturally identified with ...
12
votes
2
answers
606
views
Is there any significance to Bousfield localization in the non-derived context?
The term "Bousfield localization" of a category $C$ is used in roughly two different ways:
There is a general usage (as in model categories or triangulated categories), which $\infty$-categorically ...
5
votes
1
answer
235
views
Pontryagin square, Postnikov square and their consistency formulas
$\mathcal{P}_2$ is Pontryagin square
$$H^{2i}(M,\mathbb Z_{2^k})\to H^{4i}(M,\mathbb{Z}_{2^{k+1}}).$$
$\mathfrak{P}$ is the Postnikov square $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$
question (i)...
4
votes
1
answer
255
views
Trivialization of Pontryagin square on oriented $4$-manifolds
I'm sorry for not clearly stating my question, thanks to Robert Bruner for answering my original question, let me restate it.
Let $\mathcal{P}:H^2(-,\mathbb{Z}/2)\to H^4(-,\mathbb{Z}/4)$ be the ...
1
vote
1
answer
540
views
How to define 0-sphere in a category with zero object?
The 0-sphere $S^0$ is the coproduct of two points,
$$S^0 \simeq \ast \coprod \ast$$
How to define 0-sphere in a category with zero object?
Let $\mathcal{C}$ be a category. A cylinder, $\mathbf{I}$, ...
3
votes
1
answer
630
views
Bockstein homomorphism and Square Operations: Their consistency formulas
Here are various ways to define "Bockstein homomorphism:"
Let $\beta_p:H^*(-,\mathbb{Z}_p)
\to H^{*+1}(-,\mathbb{Z}_p)$ be the Bockstein homomorphism associated to the extension $$\mathbb{Z}...
3
votes
0
answers
173
views
Pontryagin square on spin and non-spin manifold
The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely,
$$
\mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x.
$$
...
7
votes
1
answer
382
views
Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$
The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely,
$$
\mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x.
$$
...
7
votes
3
answers
1k
views
Maps from 2-Torus to SO(3)
Could someone please point me to a reference for topologically nontrivial maps from 2-Torus to SO(3), and how they are classified? [I'm a physicist, so a simple explanation would be useful]
3
votes
2
answers
509
views
Weak homotopy equivalence between $\Omega \underset{\rightarrow}{\lim}Z_n$ and $\underset{\rightarrow}{\lim}\Omega Z_n$
Let $Z_1 \rightarrow Z_2 \rightarrow\cdots$ be an arbitrary sequence of CW-complexes and let $\Omega X$ denote the loop space over $X$. In Allen Hatcher's "Algebraic Topology" (http://pi.math.cornell....
8
votes
1
answer
734
views
Modules over Hopf Algebras and $E_2$-algebras
Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: https://arxiv.org/pdf/1309.2820.pdf
I was wondering if anybody knows of a nice relationship between ...
4
votes
1
answer
587
views
Sphere spectrum, Thom spectrum, and Madsen-Tillmann bordism spectrum
This is a following up question of Sphere spectrum, Character dual and Anderson dual.
What are the differences and the significances of the following:
(1). Homotopy classes of maps from a Thom ...
7
votes
0
answers
221
views
Does Deligne's exceptional series lead to an "exceptional K-theory"?
To a certain extent, Deligne's exceptional series $A_1 \subset A_2 \subset G_2 \subset D_4 \subset F_4 \subset E_6 \subset E_7 \subset E_8$ plays a role analogous to the classical series $A_n \subset ...
10
votes
2
answers
2k
views
Sphere spectrum, Character dual and Anderson dual
The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres.
However, could you help me to appreciate the mathematical meanings of the following:
What is the significance of ...
2
votes
0
answers
92
views
Inflation of $w_j(V_{SO(N)})$ and $w_j(M)$ from $SO(N)$ to $Spin(N)$ or Spin geometry
We know well this short exact sequence
$$
1 \to \mathbb{Z}_2 \to Spin(N) \to SO(N) \to 1.
$$
The $j$-th Stiefel-Whitney class of the associated vector bundle of $SO(N)$, as $w_j(V_{SO(N)})$, can be ...
7
votes
0
answers
167
views
Does a homological Segal condition make sense?
A $\Gamma$-space is a pointed functor from pointed finite sets to pointed spaces. Segal said that a $\Gamma$-space $F$ is special if the natural map $F(X\vee Y)\to F(X)\times F(Y)$ is a weak homotopy ...
7
votes
1
answer
460
views
Commutativity up to homotopy implies strict commutativity, for lifting problems
Suppose we have a commutative diagram
$\require{AMScd}$
\begin{CD}
A @>>> X \\
@VVV & @VVV \\
W @>>> Y\\
\end{CD}
where the map $A\rightarrow W$ is a cofibration and the ...
10
votes
1
answer
549
views
How to write K-theory Conner-Floyd Chern classes in terms of Adams operations?
From Adams, we know that the algebra of (unstable, degree-zero) cohomology operations $K^0(BU)$ can be written as formal infinite linear combinations of canonical generators
$$\mu_n := \sum_{i=0}^{n}...
18
votes
1
answer
562
views
Milnor Conjecture on Lie groups for Morava K-theory
A conjecture by Milnor state that if $G$ is a Lie group, then the map $B(G^{disc})\to BG$ sending the classifying space of $G$ endowed with the discrete topology to the classifying space of the ...
17
votes
0
answers
606
views
Which rings are cohomology rings?
Which rings can arise as cohomology rings of algebraic varieties?
To be more specific, take a Weil cohomology theory $H^*$ with coefficients in a field $K$ of characteristic 0 defined for smooth ...
5
votes
0
answers
652
views
Questions about obstruction theory (Hatcher's book)
I'm actually studying obstruction theory as presented in the last section of chapter $4$ of the book Algebraic Topology by Allen Hatcher. He first finds condition so that a space $X$ admits a ...
3
votes
1
answer
623
views
Residues and Gysin long exact for open varieties
I am familiar with the following:
let $X$ be a smooth projective complex variety, $D$ a smooth divisor in $X$ and $U=X \setminus Z$. Then there is on the one hand a residue map
$$
\mathrm{Res}_D \...
8
votes
1
answer
432
views
Spin cobordism v.s. KO theory in low or in any dimensions
It seems that from this webpage, the spin cobordism is equivalent to KO theory in low dimension.
If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$
...
3
votes
1
answer
488
views
A question on eversion of (odd) spheres
At the right column of the page 654 of the paper,
R. Palais, The Visualization of Mathematics: Towards a mathematical Exploratorium, Notice AMS it is written "There can be no eversion of ...
5
votes
0
answers
89
views
Morse theory for pairs of submanifolds of complementary dimension
If you have a closed monotone symplectic manifold $M$, then to any pair of closed monotone Lagrangian submanifolds $L_1$, $L_2$ you can associate (modulo some bubbling assumptions) a $\mathbb{Z}_N$-...
3
votes
0
answers
159
views
Reference for specific detail on Serre spectral sequence
In "A primer on spectral sequences" http://www.math.uchicago.edu/~may/MISC/SpecSeqPrimer.pdf (by J.P.May apparently, although no name is given in the pdf) I found a very detailed version of the ...
9
votes
0
answers
129
views
Relating bordism groups of $\Omega_{d}^{Spin_c}$ and $\Omega_{d}^{(Spin \times SU(N))/\mathbb{Z}_2}$ to that of $U(N)$
I felt that the earlier question may be too challenging, so let me provide a different angle and more infos to tackle an easier and separate problem.
Let us consider a more explicit a short exact ...