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

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The multiplication on $THH$ of finite fields

Let $k$ be a finite field, $THH(k)$ its topological Hochschild homology spectrum. For essentially formal reasons, we know that it's an $E_\infty$-algebra over the Eilenberg-Mac Lane spectrum $Hk$, and ...
7
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1answer
108 views

K-groups of a permutative category - are they finite?

Let $\mathcal C$ be a permutative category, that is a symmetrical monoidal category with strict associativity. One can then define the $K$-groups of $\mathcal C$, for $n >0$ by $$K_n(\mathcal C) = ...
7
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108 views

Alexander duality for non-manifolds

Let $X$ be a CW complex and $A$ a subcomplex. I will assume that both are compact, and that $X$ is $n$-dimensional. Furthermore, assume that the local homology of $X$ is that of a manifold in a ...
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88 views

Inverse limit in shape theory

Is the shape theory of Hausdorff compact spaces complete with respect to the inverse limit operation?--complete means that for every inverse system of Hausdorff compact spaces, and the shape morphisms ...
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210 views

Why does $Mf$ always support an $Mf$-orientation?

Let $f:X\to BGL_1(\mathbb{S})$ be a morphism of $E_n$-spaces and determine a principle $GL_1(\mathbb{S})$-bundle over $X$. Then it can be shown in the classical case that there is always a Thom ...
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185 views

What is the cokernel of the map $H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow H^2(X,\mathbb Z).$

For a manifold $X$ (for simplicity, assumed to be compact), let $\pi_1(X)$ be the fundamental group of $X$. What is the cokernel of the map $$H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow ...
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206 views

When are configuration spaces aspherical?

It is a theorem of Fox and Neuwirth that the space $C_k \mathbb R^2$ of unordered configurations of $k$ points in $\mathbb R^2$ is apsherical, i.e. has trivial higher homotopy groups. This has some ...
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154 views

Homotopy transfer in the opposite direction

Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...
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1answer
156 views

Generalization of Borsuk-Ulam to arbitrary ratio

Let $g: S^n \to R^n$ be a continuous odd function (i.e. $g(-x)=-g(x)$ for all $x$). The Borsuk-Ulam theorem implies that $g$ has a zero, i.e. there is an $x$ such that $g(x)=(0,0,...,0)$. Suppose $g$ ...
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131 views

Fixed point property for intersection of spaces which are homeomorphic to a disk

The following question is question 9.8 from Miller's paper ``Some interesting problems '': Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every $n\in \omega, ...
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When can a class in $H^1(M;\mathbb{Z})$ be represented by a fiber bundle over $S^1$

For a topological space M, It is known from homotopy theory that the elements of the first cohomology $H^1(M;\mathbb{Z})$ are in 1-1 correspondence with homotopy classes of maps $[M,S^1]$ In my case ...
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121 views

Hochschild-Serre spectral sequence

The Hochschild-Serre spectral sequence says that for a short exact sequence $$1 \to G \to H \to K \to 1 \quad (1)$$ of (discrete) groups, we have a first quadrant spectral sequence with $E_2$ page ...
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1answer
82 views

If $\cup (A_i \cup -A_i)=\mathbb S^d$, then is there an $x$ in $d$ sets?

If for some collection of open sets $\cup_{i\in I} (A_i \cup^* -A_i)=\mathbb S^d$, then is there an $x\in \mathbb S^d$ and $i_1,\ldots i_d\in I$ for which $x\in A_{i_1}\cap \ldots \cap A_{i_d}$? ...
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166 views

May's infinite loop machine for Friedlander's result for Adams conjecture

Eric M. Friedlander in the paper The infinite loop Adams conjecture via Classification Theorem for $\Gamma$-spaces proved the infinite loop Adams conjecture using techniques involved $\Gamma$-space. ...
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2answers
255 views

The classifying space of an infinite totally ordered set is contractible

I asked this question on math.stackexchange, but no one answered. Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This ...
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40 views

Looking for Information on Local Degree of Maps on Homology Manifolds

By a homology $n$-manifold, we mean a topological space $X$ such that for all $x \in X$: 1: if $k \neq n$ then $H_k(X, X-x)=0$ 2: $H_n(X,X-x) \cong \mathbb{Z}$. Given homology $n$-manifolds $X$ and ...
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75 views

Maps between equivariant loop spaces

I have an elementary question about equivariant loop spaces that I feel it should be well known. Given a finite group $G$ and a finite $G$-set $J$ let $S^J=\mathbb{R}[J]^+$ be the permutation ...
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1answer
147 views

Homology of manifold with action of group

Sorry for my ignorance in advance, this should be a very naive question and I would be happy for a reference. Let $G$ be an arbitrary group (not necessary finite) acting on two (connected) manifolds ...
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128 views

Fundamental groups of stably parallelizable manifolds

Is it possible to realize every finitely presented solvable group as a fundamental group of a stably parallelizable closed n-manifold? If not, are there any known counterexamples?
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Generalize $\pi_0(B\mathcal{C})\cong\{\text{objects}\}/\{\text{morphisms}\}$ to categories internal to topological spaces

Warmup: Let $\mathcal{C}$ be an ordinary category. Denote by $$B\mathcal{C}=(\coprod_{i\in\mathbb{N_0}}N_{i}(\mathcal{C})\times\Delta^i)/\tilde{}$$ its classifying space, i.e. the geometric ...
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175 views

Kernel of flux homomorphism (Calabi invariant) for volume-preserving maps on a compact manifold

Good morning everybody, I am currently reading through the book of Banyaga "Structure of classical diffeomorphism groups" link, and I am particularly interested in the question of factorizing ...
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1answer
77 views

Loop space of Fredholm operators from a Relative loop space

Atiyah and Singer proved that the nontrivial component of the set of skew-adjoint Fredholm operators $ \hat{\mathcal{F}_{*}}(\mathscr{H})$ is homotopic to the loop space of Fredholm operators ...
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107 views

Representations and K-theory of a finite group

This question is motivated by the calculation of the higher algebraic $K$-groups of a finite field. Let $G$ be a finite group, the case I am most interested in is $G = \text{Gl}_n(\mathbb F_q)$, but ...
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82 views

unordered configuration space of pointed space

Let $(X,*)$ be a pointed topological space. Let $F(X,k)=\{(x_1,\cdots,x_k)\in X^k\mid x_i\neq x_j, i\neq j\}$. Let $F(X,k)/S_k$ be the $k$-th unordered configuration space. Is there an inclusion ...
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4answers
599 views

Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?

That singular and de Rham cohomologies of a smooth manifold are isomorphic has two proofs that I know of. The classical one uses Stokes' theorem to give the isomorphism explicitly. The second proof ...
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1answer
194 views

Homotopy type of embeddings of circle in the plane

What is the homotopy type of the space of (topological) emdeddings of $S^1$ in $\mathbb R^2$? My conjecture: This space deformation retracts to $S^1\sqcup S^1$, and a retraction in each of ...
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165 views

Loop space of manifold [closed]

Question A: The free loop space of a manifold is also a manifold? Question B: The free loop space of an algebraic variety is also a algebraic variety ? Are these questions asked or answered anywhere ...
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125 views

Is the suspension of a weak equivalence again a weak equivalence?

Of course, the answer to this question depends on what we mean by suspension. If we work with based spaces and take the reduced suspension, the answer seems to be NO: Take $X = \mathbb N$ (a ...
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1answer
112 views

Unordered configuration space of $\mathbb{R}P^1$

In the paper GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2): 159-165, Theorem 2. (b): ...
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Do there exist nontrivial motivic cohomology operations preserving weights?

Suppose that for each field $F$ a linear map $X(F): H_M^{p,q}(F, \mathbb{Q}) \longrightarrow H_M^{p,q}(F,\mathbb{Q})$ is given, such that $X$ commutes with inclusions of fields and transfers for ...
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1answer
173 views

Multiplicative structure in the cohomological Leray-Serre spectral sequence - please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
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1answer
166 views

Singular homology of the zero loci of polynomials

I am very sorry but apparently I am really weak in cohomology flavored questions. I try to reformulate my problem in a very simple and hopefully clear way. This question is related with a problem in ...
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1answer
181 views

Contractible and Delta-generated implies strong deformation retract to a point?

If a CW-complex is contractible, then it strongly deformation retracts onto the inclusion of a point. However for general spaces it is well-known that just because a space is contractible, it does ...
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1answer
289 views

Action of $\mathbb{CP}^\infty$ on $U(\infty)$

For a finite CW-complex $X$, the K-theory group $K^{-1}(X)$ is isomorphic to the group of homotopy classes of maps $[X, U(\infty)]$. The group of isomorphism classes of line bundles on $X$, which I ...
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267 views

cup-length of the first Chern class of complex grassmannian

Let $G_2(\mathbb{C}^{n+1})$ be the complex grassmannian. Then the cohomology ring $H^*(G_2(\mathbb{C}^{n+1});\mathbb{C})=\mathbb{C}[c_1,c_2]/(f_n,f_{n+1})$, where ...
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134 views

Reference request: Flipping the factors in the Künneth formula

I would like to know if there is a reference for the fact that the following diagram commutes: $$ \begin{array}{ccccccccc} 0 & \to & H_*(X) \otimes H_*(Y) & \to & H_*(X\times Y) & ...
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Can stable stems be generated by homotopy operations?

The motivation for this question comes from J. Cohen's result; at the prime $p=2$ his result says that any element in ${_2\pi_*^s}$ can be written as a (higher) Toda bracket of $2,\eta,\nu,\sigma$, ...
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200 views

Serre Spectral Sequence and Cohomology Ring of Circle Bundles

I have the following (maybe simple) question about the cup product structure in the Serre spectral sequence. Consider a fiber bundle $S^1 \rightarrow E \rightarrow B$ with euler class $e \in H^2(B)$. ...
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1answer
181 views

Dyer-Lashof operations and the homology of GL_n

For any ring R, $\bigsqcup_n {BGL}_n(R)$ is an $E_\infty$-space. Are there examples of rings where people have calculated $H_*(\bigsqcup_n {BGL}_n(R);\mathbb{Z}/2)$ and determined the Dyer-Lashof ...
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1answer
107 views

LES for relative cohomology via sheaves

I was unable to find a suitable answer for the following question: Once one learns that singular cohomology is the same as cohomology with coefficients in locally constant sheaf, it is natural to try ...
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2answers
384 views

Exact sequences of pointed sets - two definitions

It seems to me that there are (at least) two notions of exact sequences in a category: 1) Let $\mathcal{C}$ be a pointed category with kernels and images. Then we call a complex (i.e. the composite ...
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985 views

Combinatorics of K(Z,2)?

Anybody knows a semi-simplicial model for $K(Z,2)$ having finite number of simplexes in any dimension? With some regular description? I have heard about big activity on triangulating $CP^n$ but this ...
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1answer
168 views

Classifying space of a colimit of topological categories

Say I have a diagram $D:I\rightarrow\text{Cat}(\text{Top})$ of categories internal to compactly generated topological spaces. This induces a diagram $BD:I\rightarrow \text{Top}$ of classifying spaces. ...
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Why should have Peter May worked with CGWH instead of CGH in “The Geometry of Iterated Loop Space”?

This is a follow-up to Dan Ramras' answer of this question. The following correction can be found in the errata to The Geometry of Iterated Loop space (Page 484 here). The weak Hausdorff rather ...
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Lattice-isotopic essentialization of arrangements

I'm working on a problem related to $\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...
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127 views

A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$. The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence $$ ...
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235 views

When are principal bundles preserved by colimits?

Let $G$ be a topological group and consider a family $$G\rightarrow E_i\rightarrow B_i$$ of $G$-principal bundles indexed over the natural numbers. Suppose we have $G$-bundle morphisms (equivariant ...
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67 views

first chern class versus compactifying divisor in Ramanujam's surface

I have an elementary question about Ramanujam's surface. Ramanujam's surface is naturally the complement of a singular divisor $D$ in the one point blow up of $CP^2$, $\mathbb{F}_1$. One can resolve ...
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1answer
230 views

A survey for various $K$-homology theories and their relationship

The ordinary Topological $K$ theory defined by Atiyah and Hirzebruch is a generalized cohomology theory (see wikipedia).There is the Bott spectrum associated to this generalized cohomology ...
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1answer
234 views

A question on Hawaiian earring

I have asked this question in MSE but have not got any satisfactory answer, so I am asking it here. Any idea on how to approach this problem will be highly appreciated. Consider the Hawaiian earring. ...