End extension

In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex X (more geometrically a Poincaré space), a normal map on X endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, X has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold M to X matching the fundamental classes and preserving normal bundle information. If the dimension of X is $\ge$ 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to X actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov.

The cobordism classes of normal maps on X are called normal invariants. Depending on the category of manifolds (differentiable, piecewise-linear, or topological), there are similarly defined, but inequivalent, concepts of normal maps and normal invariants.

It is possible to perform surgery on normal maps, meaning surgery on the domain manifold, and preserving the map. Surgery on normal maps allows one to systematically kill elements in the relative homotopy groups by representing them as embeddings with trivial normal bundle.

Definition

There are two equivalent definitions of normal maps, depending on whether one uses normal bundles or tangent bundles of manifolds. Hence it is possible to switch between the definitions which turns out to be quite convenient.

1. Given a Poincaré complex X (i.e. a CW-complex whose cellular chain complex satisfies Poincaré duality) of formal dimension $n$, a normal map on X consists of

• a map $f:M\to X$ from some closed n-dimensional manifold M,
• a bundle $\xi$ over X, and a stable map from the stable normal bundle ${\nu }_M$ of $M$ to $\xi$, and
• usually the normal map is supposed to be of degree one. That means that the fundamental class of $M$ should be mapped under $f$ to the fundamental class of $X$: $f_{*}([M])=[X]\in H_n(X)$.

2. Given a Poincaré complex $X$ (i.e. a CW-complex whose cellular chain complex satisfies Poincaré duality) of formal dimension $n$, a normal map on $X$ (with respect to the tangent bundle) consists of

• a map $f:M\to X$ from some closed $n$-dimensional manifold $M$,
• a bundle $\xi$ over $X$, and a stable map from the stable tangent bundle ${\tau }_M\oplus {\epsilon }^k$ of $M$ to $\xi$, and
• similarly as above it is required that the fundamental class of $M$ should be mapped under $f$ to the fundamental class of $X$: $f_{*}([M])=[X]\in H_n(X)$.

Two normal maps are equivalent if there exists a normal bordism between them.

Role in surgery theory

Surgery on maps versus surgery on normal maps

Consider the question:

Is the Poincaré complex X of formal dimension n homotopy-equivalent to a closed n-manifold?

A naive surgery approach to this question would be: start with some map $M\to X$ from some manifold $M$ to $X$, and try to do surgery on it to make a homotopy equivalence out of it. Notice the following: Since our starting map was arbitrarily chosen, and surgery always produces cobordant maps, this procedure has to be performed (in the worst case) for all cobordism classes of maps $M\to X$. This kind of cobordism theory is a homology theory whose coefficients have been calculated by Thom: therefore the cobordism classes of such maps are computable at least in theory for all spaces $X$.

However, it turns out that it is very difficult to decide whether it is possible to make a homotopy equivalence out of the map by means of surgery, whereas the same question is much easier when the map comes with the extra structure of a normal map. Therefore, in the classical surgery approach to our question, one starts with a normal map $f:M\to X$ (suppose there exists any), and performs surgery on it. This has several advantages:

• The map being of degree one implies that the homology of $M$ splits as a direct sum of the homology of $X$ and the so-called surgery kernel $K_{*}(M)=ker(f_{*}:H_{*}(M)\to H_{*}(X))$, that is $H_{*}(M)=K_{*}(M)\oplus H_{*}(X)$. (Here we suppose that $f$ induces an isomorphism of fundamental groups and use homology with local coefficients in $Z[{\pi }_1(X)]$.)

By Whitehead's theorem, the map $f$ is a homotopy equivalence if and only if the surgery kernel is zero.

• The bundle data implies the following: Suppose that an element $\alpha \in {\pi }_{p+1}(f)$ (the relative homotopy group of $f$) can be represented by an embedding $\varphi :S^p\to M$ (or more generally an immersionTemplate:Disambiguation needed) with a null-homotopy of $f\circ \varphi :S^p\to X$. Then it can be represented by an embedding (or immersion) whose normal bundle is stably trivial. This observation is important since surgery is only possible on embeddings with a trivial normal bundle. For example, if $p$ is less than half the dimension of $X$, every map $S^p\to X$ is homotopic to an embedding by a theorem of Whitney. On the other hand, every stably trivial normal bundle of such an embedding is automatically trivial, since ${\pi }_p(BO,B{O}_k)=0$ for $k>p$. Therefore, surgery on normal maps can always be done below the middle dimension. This is not true for arbitrary maps.

Notice that this new approach makes it necessary to classify the bordism classes of normal maps, which are the normal invariants. Contrarily to cobordism classes of maps, the normal invariants are a cohomology theory. Its coefficients are known in the case of topological manifolds. For the case of smooth manifolds, the coefficients of the theory are much more complicated.

Normal invariants versus structure set

There are two reasons why it is important to study the set $\mathcal{𝒩}(X)$. Recall that the main goal of surgery theory is to answer the questions:

1. Given a finite Poincaré complex $X$ is there an $n$-manifold homotopy equivalent to $X$?

2. Given two homotopy equivalences $f_i:M_i\to X$, where $i=0,1$ is there a diffeomorphism $h:M_0\to M_1$ such that $f_1\circ h\simeq f_0$?

Notice that if the answer to these questions should be positive then it is a necessary condition that the answer to the following two questions is positive

1.' Given a finite Poincaré complex $X$ is there a degree one normal map $(f,b):M\to X$?

2.' Given two homotopy equivalences $f_i:M_i\to X$, where $i=0,1$ is there a normal cobordism $(F,B):(W,M,M^{\prime })\to (M\times I,M\times 0,M\times 1)$ such that $\partial F_0=f_0$ and $\partial F_1=f_1$?

This is of course an almost trivial observation, but it is important because it turns out that there is an effective theory which answers question 1.' and also an effective theory which answers question 1. provided the answer to 1.' is yes. Similarly for questions 2. and 2.' Notice also that we can phrase the questions as follows:

1.' Is $\mathcal{𝒩}(X)\ne \mathrm{\varnothing }$?

2.' Is $f_0=f_1$ in $\mathcal{𝒩}(X)$?

Hence studying $\mathcal{𝒩}(X)$ is really a first step in trying to understand the surgery structure set $\mathcal{𝒮}(X)$ which is the main goal in surgery theory. The point is that $\mathcal{𝒩}(X)$ is much more accessible from the point of view of algebraic topology as is explained below.

Homotopy theory

1.' Let X be a finite n-dimensional Poincaré complex. It is useful to use the definition of $\mathcal{𝒩}(X)$ with normal bundles. Recall that a (smooth) manifold has a unique tangent bundle and a unique stable normal bundle. But a finite Poincaré complex does not possess such a unique bundle. Nevertheless, it possesses a substitute - a unique in some sense spherical fibration - the so-called Spivak normal fibration. This has a property that if $X$ is homotopy equivalent to a manifold then the spherical fibration associated to the pullback of the normal bundle of that manifold is isomorphic to the Spivak normal fibration. So it follows that if $\mathcal{𝒩}(X)\ne \mathrm{\varnothing }$ then the Spivak normal fibration has a bundle reduction. By the Pontrjagin-Thom construction the converse is also true.

This can be formulated in terms of homotopy theory. Recall $BG$ the classifying space for stable spherical fibrations, $BO$ the classifying space for stable vector bundles and the map $J:BO\to BG$ which is induced by the inclusion $O\hookrightarrow G$ and which corresponds to taking the associated spherical fibration of a vector bundle. In fact we have a fibration sequence $BO\to BG\to B(G/O)$. The Spivak normal fibration is classified by a map ${\nu }_X:X\to BG$. It has a vector bundle reduction if and only if ${\nu }_X$ has a lift ${\tilde{\nu }}_X:X\to BO$. This is equivalent to requiring that the composition $X\to BG\to B(G/O)$ is null-homotopic.

Note that the homotopy groups of $B(G/O)$ are known in certain low-dimensions and are non-trivial which suggests the possibility that the above condition can fail for some $X$. There are in fact such finite Poincaré complexes, and the first example was obtained by Gitler and Stasheff,Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. yielding thus an example of a Poincaré complex not homotopy equivalent to a manifold.

2.' Relativizing the above considerations one obtains an (unnatural) bijection

$\mathcal{𝒩}(X)\cong [X,G/O].$

Different categories

The above bijection gives $\mathcal{𝒩}(X)$ a structure of an abelian group since the space $G/O$ is a loop space and in fact an infinite loop space so the normal invariants are a zeroth cohomology group of an extraordinary cohomology theory defined by that inifinite loop space. Note that similar ideas apply in the other categories of manifolds and one has bijections

$\mathcal{𝒩}(X)\cong [X,G/O]$, and ${\mathcal{𝒩}}^{PL}(X)\cong [X,G/PL]$, and ${\mathcal{𝒩}}^{TOP}(X)\cong [X,G/TOP].$

It is well known that the spaces

$G/O$, $G/PL$ and $G/TOP$

are mutually not homotopy equivalent and hence one obtains three different cohomology theories.

Sullivan analyzed the cases $G/PL$ and $G/TOP$. He showed that these spaces possess alternative inifinite loop space structures which are in fact better from the following point of view: Recall that there is a surgery obstruction map from normal invariants to the L-group. With the above described groups structure on the normal invariants this map is NOT a homomorphism. However, with the group structure from Sullivan's theorem it becomes a homomorphism in the categories $CAT=PL$, and $TOP$. His theorem also links these new group structures to the well-known cohomology theories: the singular cohomology and real K-theory.

References

• 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.
  
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  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
• 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