Heath Robinson (codebreaking machine)

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.

Motivation

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map

$\eta \colon S^{3}\to S^{2}$,

and proved that $\eta$ is essential, i.e. not homotopic to the constant map, by using the linking number (=1) of the circles

$\eta ^{-1}(x),\eta ^{-1}(y)\subset S^{3}$ for any $x\neq y\in S^{2}$.

It was later shown that the homotopy group $\pi _{3}(S^{2})$ is the infinite cyclic group generated by $\eta$. In 1951, Jean-Pierre Serre proved that the rational homotopy groups

$\pi _{i}(S^{n})\otimes \mathbb {Q}$

for an odd-dimensional sphere ($n$ odd) are zero unless i = 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree $2n-1$. There is an interesting way of seeing this:

Definition

Let $\phi \colon S^{2n-1}\to S^{n}$ be a continuous map (assume $n>1$). Then we can form the cell complex

$C_{\phi }=S^{n}\cup _{\phi }D^{2n},$

where $D^{2n}$ is a $2n$-dimensional disc attached to $S^{n}$ via $\phi$. The cellular chain groups $C_{\mathrm {cell} }^{*}(C_{\phi })$ are just freely generated on the $n$-cells in degree $n$, so they are $\mathbb {Z}$ in degree 0, $n$ and $2n$ and zero everywhere else. Cellular (co-)homology is the (co-)homology of this chain complex, and since all boundary homomorphisms must be zero (recall that $n>1$), the cohomology is

$H_{\mathrm {cell} }^{i}(C_{\phi })={\begin{cases}\mathbb {Z} &i=0,n,2n,\\0&{\mbox{otherwise}}.\end{cases}}$

Denote the generators of the cohomology groups by

$H^{n}(C_{\phi })=\langle \alpha \rangle$ and $H^{2n}(C_{\phi })=\langle \beta \rangle .$

For dimensional reasons, all cup-products between those classes must be trivial apart from $\alpha \smile \alpha$. Thus, as a ring, the cohomology is

$H^{*}(C_{\phi })=\mathbb {Z} [\alpha ,\beta ]/\langle \beta \smile \beta =\alpha \smile \beta =0,\alpha \smile \alpha =h(\phi )\beta \rangle .$

The integer $h(\phi )$ is the Hopf invariant of the map $\phi$.

Properties

Theorem: $h\colon \pi _{2n-1}(S^{n})\to \mathbb {Z}$ is a homomorphism. Moreover, if $n$ is even, $h$ maps onto $2\mathbb {Z}$.

The Hopf invariant is $1$ for the Hopf maps (where $n=1,2,4,8$, corresponding to the real division algebras $\mathbb {A} =\mathbb {R} ,\mathbb {C} ,\mathbb {H} ,\mathbb {O}$, respectively, and to the double cover $S(\mathbb {A} ^{2})\to \mathbb {PA} ^{1}$ sending a direction on the sphere to the subspace it spans). It is a theorem, proved first by Frank Adams and subsequently by Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1.

Generalisations for stable maps

A very general notion of the Hopf invariant can be defined, but it requires a certain amount of homotopy theoretic groundwork:

Let $V$ denote a vector space and $V^{\infty }$ its one-point compactification, i.e. $V\cong \mathbb {R} ^{k}$ and

$V^{\infty }\cong S^{k}$ for some $k$.

If $(X,x_{0})$ is any pointed space (as it is implicitly in the previous section), and if we take the point at infinity to be the basepoint of $V^{\infty }$, then we can form the wedge products

$V^{\infty }\wedge X$.

Now let

$F\colon V^{\infty }\wedge X\to V^{\infty }\wedge Y$

be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of $F$ is

$h(F)\in \{X,Y\wedge Y\}_{\mathbb {Z} _{2}}$,

an element of the stable $\mathbb {Z} _{2}$-equivariant homotopy group of maps from $X$ to $Y\wedge Y$. Here "stable" means "stable under suspension", i.e. the direct limit over $V$ (or $k$, if you will) of the ordinary, equivariant homotopy groups; and the $\mathbb {Z} _{2}$-action is the trivial action on $X$ and the flipping of the two factors on $Y\wedge Y$. If we let

$\Delta _{X}\colon X\to X\wedge X$

denote the canonical diagonal map and $I$ the identity, then the Hopf invariant is defined by the following:

$h(F):=(F\wedge F)(I\wedge \Delta _{X})-(I\wedge \Delta _{Y})(I\wedge F).$

This map is initially a map from

$V^{\infty }\wedge V^{\infty }\wedge X$ to $V^{\infty }\wedge V^{\infty }\wedge Y\wedge Y$,

but under the direct limit it becomes the advertised element of the stable homotopy $\mathbb {Z} _{2}$-equivariant group of maps. There exists also an unstable version of the Hopf invariant $h_{V}(F)$, for which one must keep track of the vector space $V$.

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
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