TITLE: Topological description of Manifold with boundary
QUESTION [5 upvotes]: Is there any 'general' topological invariant to tell the difference between $M$ and $N$, where $M$ has homotopy type of a closed manifold and $N$ has homotopy type of a manifold with boundary. 
I meant something like homotopy group/ homology group can 'detect' the obstruction of being homotopic to a closed manifold.

REPLY [4 votes]: Perhaps the fundamental algebraic difference between a compact manifold with boundary and one without boundary is whether it satisfies Poincare duality or Lefschetz duality. Indeed, in algebraic surgery theory one defines the boundary of a chain complex as a "correction term" to a "Poincare duality" which does not hold, to turn it into a Lefschetz duality which does hold. This definition is fundamental to the discipline.
In more detail, let R be a ring with involution and consider a bounded chain complex over R
$C_\bullet:=C_0\longrightarrow C_{1}\longrightarrow\cdots\longrightarrow C_n.$
The dual cochain complex is 

$C^\bullet:=C^n\longrightarrow C^{n-1}\longrightarrow\cdots\longrightarrow C^0,$
where by definition $C^i:= Hom_R(C_i,R)$.
Poincare duality PD:Hk(X)$\longrightarrow$ Hn-k(X) is induced by taking the cap product with a fundamental class $[X]\in C_n$. On a chain level, the cap product is given by the composition
$C^\bullet\overset{\Delta}{\longrightarrow }(C^\bullet)^{op}\otimes C^\bullet(X)\overset{\setminus}{\longrightarrow} Hom_R(C^\bullet,C_\bullet),$
where the first map is induced from the diagonal map via the Eilenberg-Zilber theorem, and the second is the slant map. The traditional name for $\setminus\Delta[X]$ is $\varphi_0$, which is a chain map from $C^\bullet$ to $C_\bullet$. Algebraic surgery theory also defines chain maps $\varphi_1,\ldots,\varphi_n$, which fit together to form a collection {$\varphi_0,\ldots,\varphi_n$} called a symmetric structure on the chain complex $C_\bullet$. The boundary of a chain complex with symmetric structure is defined as the mapping cone of $\varphi_0$. That's your invariant- no boundary if and only if that mapping cone is chain homotopic to zero.
This leads to definitions of L-groups, algebraic cobordism, algebraic surgery, and all kind of good and deep mathematics which goes along with all of that.
If you want to know more, the canonical reference for this sort of stuff perhaps, which has sort of a cult status, is Andrew Ranicki's ATS1.