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

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Moment maps and flat degenerations of toric varieties

We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$. How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...
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1answer
134 views

The kernel of the natural map $\pi_k(BU(r)) \to \pi_k (BU)$

Is this group known outside of the stable range? If so, what is it? If not, what is known about it?
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1answer
205 views

When does the free loop space fibration split?

This question is a repost from stack.exchange. It didn't get a lot of attention there. Perhaps it is badly written (or silly?). If so, I'd be happy to get comments/suggestions about that. Let $X$ be ...
5
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1answer
184 views

Map between homotopy groups of O, related to J-homomorphism and K-theory of Z

Let $s \geq 0$ be fixed. The $J$-homomorphism includes $\pi_{8s+1}(SO) = \mathbb Z/2$ in $\pi_{8s+1}^s$, the $(8s+1)$-th stable homotopy group of spheres. Now regard $\pi_{8s+1}^s = \pi_{8s+1} ...
32
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1answer
790 views

What can topological modular forms do for number theory?

Topological modular forms ($TMF$) have in the recent years made an impact in algebraic topology. Roughly, the spectrum $tmf$ is the (derived) global sections of the sheaf of $E_\infty$ ring spectra ...
3
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2answers
232 views

Acyclic complexes for extraordinary cohomology theories

Let $X$ be a CW complex such that for all extraordinary homology theories, if you plug $X$ into them you get the same value as plugging in a point. Must $X$ be contractible?
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101 views

Injectivity of the Dehn-Nielsen-Baer map?

If $S$ is a closed hyperbolic surface, is there an easy proof of the injectivity of the Dehn-Nielsen-Baer map from $\mathrm{Mod}(S)$ to $\mathrm{Out}(\pi_1(S))$, taking an element of the mapping class ...
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114 views

How to give a $\Delta$-complex structure?

The quotient space of a finite collection of disjoint 2-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic with $\mathbb{R^2}$. But I am not able to prove , ...
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349 views

What's the detailed proof of “the composition of planar tangles is well-defined”?

In the planar algebra theory (see here or there section 2), a planar tangle is an isotopy class; then to define the composition of two tangles, we need to choose a representative in each classes. See ...
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2answers
261 views

Is Eilenberg-Maclane $\wedge$ Moore space the spectrum of the cohomology theory $H^*(\ ,G)$?

In the web page http://www.encyclopediaofmath.org/index.php/Moore_space it can be found the following statement: If $K(\mathbb Z,n)$ is the Eilenberg–MacLane space of the group of integers ...
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971 views

When is there a submersion from a sphere into a sphere?

(First posted on math.SE, with no answers.) That is: For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$? The discussion at this math.SE question has ...
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1answer
211 views

Naive G-spectrum representing geometric equivariant cobordism

Let $G$ be a finite group. By the transversality results of Wasserman $G$-equivariant bordism (say real) should be a naive homology theory, and as such it should be represented by a naive G-spectrum. ...
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2answers
207 views

Convergence of a sum with the ranks of homotopy groups

Let $F$ be a (nontrivial) topological space that satisfies the following conditions: 1) $\pi_n(F)$ has a trivial action of $\pi_1(F)$ for $n>0$ and 2) its homology groups are finitely generated. ...
4
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0answers
57 views

Multi-podal points

Two points $x,y \in \mathbb{R}^n$ are called antipodal if $x = -y$. Stated differently, $x,y$ are antipodal if: They have the same absolute value in each of their $n$ coordinates; Each of their ...
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1answer
284 views

Simply connected CW-complex with only finitely many nontrivial homotopy and homology groups

Let $X$ be a simply connected CW-complex such that all but finitely many of its homotopy groups and its homology groups (with $\mathbb Z$ coefficients) are 0. Is $X$ then necessarily contractible? I ...
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79 views

a construction on Stiefel manifolds

Are there any references concerning the following space $V(k,N,X)$ and $U(k,N,X)$? And the cohomology of these spaces? Thanks.
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96 views

The most general set-up for tensors and connections

This is maybe a too vague question, so I will try to be as specific as possible. My question is: What is the most general set-up where one can define tensors and connections? For example, we know ...
7
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1answer
220 views

Non-orientable $6$-manifold with $H_4(M)=\mathbb{Z}/2$?

Does there exist a smooth, closed, non-orientable $6$-manifold $M$ such that $H_4(M;\mathbb{Z})=\mathbb{Z}/2$?
3
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1answer
134 views

Which automorphisms on $H_{1}(M^{3})$ are induced by homotopy equivalences?

Let $M^{3}$ be a closed orientable 3-manifold, and $\phi:H_{1}(M;\mathbb{Z})\to H_{1}(M;\mathbb{Z})$ be an automorphism of abelian groups. My question is: Is there any characterization of $\phi$ ...
12
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1answer
209 views

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 ...
10
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1answer
150 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) = ...
9
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1answer
223 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 ...
3
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0answers
104 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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0answers
234 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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1answer
205 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 ...
6
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1answer
225 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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192 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 ...
3
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1answer
170 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 $g$ ...
3
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1answer
215 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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3answers
561 views

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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143 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 ...
3
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1answer
94 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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187 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. ...
4
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2answers
292 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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0answers
43 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 ...
3
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0answers
88 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
159 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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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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244 views

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 ...
5
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1answer
194 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 ...
3
votes
1answer
82 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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127 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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1answer
96 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
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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 ...
6
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1answer
202 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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1answer
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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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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
124 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 ...
1
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1answer
192 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: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...