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

4
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23 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 $H^3$, ...
2
votes
0answers
29 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 ...
3
votes
0answers
38 views

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 ...
2
votes
1answer
80 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
1answer
240 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 ...
3
votes
1answer
93 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
0answers
82 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
0answers
72 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
0answers
51 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\...
4
votes
1answer
110 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 ...
2
votes
0answers
126 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
0answers
119 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
0answers
44 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 ...
4
votes
2answers
155 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 ...
1
vote
0answers
52 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
vote
0answers
65 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
0answers
69 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 ...
15
votes
0answers
97 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, \...
-1
votes
0answers
71 views

A generalization of Conner Conjecture

Let $G$ be a compact (abelian) totally disconnected group and $X$ be a compact $G$-space. If $X$ is $% %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion $-acyclic space (i.e. $\widetilde{...
6
votes
0answers
168 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 ...
11
votes
2answers
282 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
1answer
107 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 $$H^2(M,\mathbb Z_3)\to H^5(M,\mathbb Z_9).$$ ...
4
votes
1answer
223 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 ...
0
votes
1answer
206 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
1answer
99 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}_p\to\...
2
votes
0answers
87 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, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
6
votes
1answer
232 views
+50

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, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
2
votes
2answers
267 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
2answers
158 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....
5
votes
1answer
169 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 ...
0
votes
0answers
76 views

$(X \wedge K)/(A \wedge K)= (X/A) \wedge K$

I have to prove that the following equality $$(X \wedge K)/(A \wedge K)= (X/A) \wedge K$$ holds for a CW-pair $(X, A)$ and a fixed CW-complex $K$; here $\wedge$ denotes the smash product, defined as $...
3
votes
0answers
89 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
0answers
142 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 ...
6
votes
2answers
399 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
0answers
80 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
0answers
108 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 ...
6
votes
1answer
234 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 ...
6
votes
1answer
224 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}...
16
votes
1answer
275 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
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0answers
420 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 ...
4
votes
0answers
329 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 ...
2
votes
1answer
120 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 \...
6
votes
1answer
180 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 $n$) $$\Omega_d(BG)_p.$$ ...
2
votes
1answer
190 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
0answers
52 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
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0answers
121 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
0answers
90 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 ...
6
votes
0answers
94 views

Bordism groups and a short exact sequence

Let us consider a short exact sequence: $$ 1\to N\to G\to Q \to 1, $$ where $N$, $Q$, and $G$ can be continuous Lie groups in general (or finite groups). Suppose I have the data and the computations ...
1
vote
2answers
104 views

Why is the flat cotorsion pair actually a cotorsion pair?

I asked this question some while ago on Stack Exchange but didn't get an answer (link), so I am trying it here as well. Fix a ringed space $(X,\mathcal{O})$ and denote by $\mathcal{F}$ the class of ...
4
votes
0answers
89 views

Is there a simple algebraic setup to accomodate fibres and cofibres at the same time?

If I understand it correctly, there are two mutually dual "leading principles" in homotopy theory: never perform quotients, add structure instead; never require subobjects, take fibres instead. ...