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5
votes
1answer
149 views

Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map $f:\mathbb{C}P^2\...
0
votes
0answers
37 views

Minimize a function to learn a mapping

I have two questions. I want to learn a mapping $M$ that minimizes the right-hand side of the following equation: $E =\frac{1}{N} \sum\limits_{i=1}^N \bigg(\sum\limits_{j=1}^K \alpha_i - \big(\...
3
votes
1answer
94 views

Homotopy equivalence of maps with compact support and maps which vanish at infinity

Let $X$ be a nice space, maybe a manifold, and let $Y$ be a based space. What sort of conditions must we impose on $Y$ (and $X$ if need be) to get a homotopy equivalence $$ \mathcal C_c(X,Y) \simeq \...
6
votes
1answer
177 views

cohomology of iterated loop space on spheres

In the book The homology of iterated loop spaces, the homology Hopf algebra (1) $$ H_*(\Omega^n \Sigma^n X;\mathbb{Z}_p) $$ for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the ...
2
votes
0answers
77 views

cohomology ring of mapping spaces

In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring $$ H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p) $$ is obtained for any ...
1
vote
0answers
28 views

How to rewrite this logarithmic update rule [closed]

I tried to rewrite the equation given below. I get stuck getting rid of the $ P(n|z_{1:t})$ on the left side. How can this be done? $$ P(n|z_{1:t}) = \left[1+ \frac{1-P(n|z_{t})}{P(n|z_t)} \frac{1-P(...
1
vote
1answer
356 views

When is the inclusion of a relative mapping space into a mapping space a cofibration?

Let $(X,A)$ and $(Y,B)$ be pairs of spaces and subspaces, let $\operatorname{Map}(X,Y)$ the space of maps $f:X\to Y$ equipped with the compact-open topology and let $\operatorname{Map}(X,A;Y,B)$ be ...
15
votes
1answer
840 views

Whitehead product with identity on homotopy groups of spheres

For $n\geq 2$ let $(S^n,p)$ be the $n$-sphere with a base point $p$. Let $1:S^n\to S^n$ denote the identity map. Let us define the map $Wh_1: \pi_i(S^n,p)\to \pi_{i+n-1}(S^n,p), \alpha\mapsto [\...
11
votes
2answers
1k views

Induced map on path manifolds: is it a submersion?

Consider the following claim: Let $p:M \to N$ be a (surjective) submersion of finite-dimensional smooth manifolds. Let $J$ denote one of $[0,1],\ [0,1),\ (0,1]$. Then $p_*:M^J \to N^J$ is a ...
0
votes
1answer
533 views

Remap FFT frequency bin distribution

I've coded up the FFT for a dataset I'm working with. My intent is to create a waterfall plot of the result, but the problem I'm running into is if I change my input data size, then I get a different ...
7
votes
1answer
2k views

Is there a manifold structure on a space of conformal maps?

I would be very grateful for any information or pointers for the following: 1) Fix an open subset $U$ of $\mathbb{CP}^1$. a) Does the set of all holomorphic maps from $U$ to $\mathbb{C}$ (with the ...
4
votes
1answer
161 views

Contractible space of maps between Eilenberg-Mac Lane spaces, 2

Let $G$ and $H$ be torsion abelian groups. Are the following are equivalent: $\mathrm{Hom}(G, H) = 0$ $\mathrm{map}_*( K(G, m), K(H,n)) \sim *$ for all $n, m\geq 1$ ? Clearly (2) implies (1).
4
votes
1answer
308 views

Contractible space of maps between Eilenberg-Mac Lane spaces

Suppose $A$ is an abelian torsion group, with no elements of order $p$, and let $P$ be an abelian $p$-group (i.e., the order of each element is a power of $p$). It sure seems to me that $$ \mathrm{...
6
votes
0answers
315 views

The Space of Cellular Maps

Let $X$ and $Y$ be CW complexes. Inside of the space of maps $\mathrm{map}(X,Y)$, we have the subspace $\mathrm{CW}(X,Y)$, consisting of just the cellular maps from $X$ to $Y$. The Cellular ...