050005 2013 J. J. Niemela V. Lakshminarayanan, Waterloo University, Canada
050005

We report a series of experiments on laser pulsed photoacoustic excitation in
turbid polymer samples addressed to evaluate the sound speed in the samples
and the presence of inhomogeneities in the bulk. We describe a system which
allows the direct measurement of the speed of the detected waves by engraving
the surface of the piece under study with a fiduciary pattern of black lines.
We also describe how this pattern helps to enhance the sensitivity for the
detection of an inhomogeneity in the bulk. These two facts are useful for
studies in soft matter systems including, perhaps, biological samples. We have
performed an experimental analysis on Grilon®samples in different situations
and we show the limitations of the method.

# Enhancement of photoacoustic detection of inhomogeneities in polymers

P. Grondona  [inst1] H. O. Di Rocco  [inst2] D. I. Iriarte  [inst2] J. A.
Pomarico  [inst2]
H. F. Ranea-Sandoval  [inst2] G. M. Bilmes[inst3] E-mail:
<EMAIL_ADDRESS>
(7 December 2012; 19 June 2013)

††volume: 5

99 inst1 Universidad Nacional de Rosario. Facultad de Ciencias Bioquímicas y
Farmacéuticas. Rosario (Santa Fe) Argentina. inst2 Instituto de Física “Arroyo
Seco”, Universidad Nacional del Centro de la Provincia de Buenos Aires. Calle
Pinto 399, B7000GHG, Tandil (Buenos Aires) Argentina. inst3 Centro de
Investigaciones Ópticas (CONICET-CIC) and Facultad de Ingeniería Universidad
Nacional de La Plata, La Plata. Argentina.

##  1 Introduction

In highly light-scattering materials, such as certain types of polymers,
turbid liquids, glassy structures, and body organs, inspection and monitoring
of internal features were made possible by means of X-Ray irradiation until
the development of ultrasound imaging. The former has the well-known
disadvantage that in biological tissues it may trigger degenerative processes
in the cells, and in non-biological samples, X-Ray inspection is not always
simple to perform directly in the production line. Ultrasound imaging is very
helpful in these situations.

On the other hand, visible light optical tomography and optical topography is
nowadays reaching the status of clinical resource in the detection and
monitoring of several types of tumors and for non-invasive evaluation of
oxygenation of tissues in biological samples. In non-clinical applications it
can be used for the detection of abnormal bodies within materials, which is of
great importance in quality control in several areas of technology. These
techniques were derived from the study of light propagating in turbid media,
and applied afterward to biological samples and medical imaging of different
parameters often using polymers as phantoms of biological tissues [1, 2, 3, 4,
5, 6, 7].

The photoacustic effect (PA) provides a method of analysis that has been used
in clear fluids and has sufficiently proven its capability for detecting very
low concentrations of absorbing species in a mixture or solution; it has also
been used for the monitoring of molecular processes in different environments
as shown in references [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. This paper
intends to make a contribution on the application of the PA in soft matter,
namely the detection of inclusions in polymer samples and the direct
determination of the speed of sound in the material used for the samples.

The PA technique has the advantage that acoustic waves do not scatter as light
does in the characteristic lengths of many experimental situations. Even if
the excitation light undergoes scattering, the location of an inhomogeneity
within the bulk of the sample can be achieved by detecting the remnant of the
shock wave generated at an absorbing region or at an interface at which the
speed of the sound waves changes. Repeating this inspection at other relative
positions of the laser and the acoustic detector and with the aid of a
suitable algorithm, a sufficiently precise location of a single inhomogeneity
of simple geometry can thus, in principle, be resolved, together with some
information about its composition (using at least two wavelength for the
excitation), provided the speed of sound is known (see, for example, Ref.
[20]). An example of this is presented in Ref. [21] in a rear-detection scheme
used for detecting inhomogeneities in subsurface inhomogeneities in metals.

The PA detection of bodies included in a turbid medium may provide
complementary information to diffuse light propagation studies in that medium.
Namely, it could bring an independent value for the absorption coefficient,
and it thus may help in the solution of the inverse problem in optical
tomography of samples.

The speed of sound determination relies on the fact that the acoustic signal
picked by the transducer arrives at times proportional to the distance from
the laser beam that generates the shock wave to that transducer. A drawback
with the photoacustic method applied to turbid materials is that the light
scattered by the bulk generates a pressure pulse on the detector if it is in
contact with a free surface of the sample explored. Consequently, the time of
arrival of the pressure signal at the detector is insensitive to the relative
position of the laser and the sensor. Hence, the speed of the waves involved
in the PA signal is difficult to determine and requires an adequate procedure
to evaluate it. This is one of the motivations of this contribution.

For this paper, we used a laser pulsed photoacustic system equipped with a PZT
in contact with the sample made of polymer Grilon®which is representative of a
turbid medium, to show how the presence of controlled, fiduciary absorbing
regions in the surface of a sample are used as local wave generators that
allow the determination of the speed of waves in materials despite of the
light scattering described. We have engraved in the surface of the samples a
pattern of stripes of absorbing material. In this way, we have a greater
signal whose contribution may be discriminated from the signal generated by
the light scattered by the sample material. We demonstrate that this fiduciary
pattern is useful also to enhance the photoacustic signal, and that from that
signal the presence of inhomogeneities in a medium may be inferred.

Other successful recent approaches to the problem of detection of tumoral
tissues in biological samples can be found in Refs. [18, 19].

##  2 Experimental

A scheme of the pulsed photoacustic system used in all the experiments is
shown in Fig. 1, which is essentially the same that can be used to determine
the speed of sound in liquids and in clear samples.

Figure 1: Experimental setup of the experiment. The parts are: the Nd+3:YAG
laser (NdL), the positioning device (X-TS), the Pinhole (PH) to clip the laser
beam (LB), and the oscilloscope (DO), the amplifier with a DC power supply
(APS). The DO is synchronized with the laser via the laser pulse
synchronization (LPS) cable.

We used a pulsed Nd+3:YAG laser emitting at $1.06\,\mu$m with a pulse duration
of approximately $10$ ns, at energies between $0.5$ mJ to 50 mJ. The laser
beam was clipped by means of a pinhole in order to reduce the original laser
beam size and to use a uniform spot thus reducing the power impinging on the
samples. This pinhole was held at the far end of a beam dump for security
reasons. In the results we present here, we have used two pinholes of $1$ mm
and $1.5$ mm in diameter which shall be specified in each experiment. This is
the diameter of the laser impinging on the sample, as inferred from sensitive
photographic paper. For acoustic detection, a ceramic $4\times 4$ mm2 PZT
transducer was strongly pressed against one of the free surfaces of the
sample, namely the one normal to that facing the laser. The photoacustic
signals were amplified and processed by means of a Tektronix TDS 3032B, $300$
MHz digital oscilloscope, averaging at least 64 signals before displaying the
photoacustic signal.

Samples used were square-section parallelepipeds, $10$ mm width, and $39$ mm
high, all made from the same polymer Grilon®piece. The samples were placed in
a C-clamp, with the PZT cage in one of its arms. The fiduciary pattern
engraved on one of the faces of some samples consists of five grooves of
approximately $1$ mm width and $0.2$ mm deep, filled with thick black paint,
separated by stripes of material which retain the natural turbid white color
of the polymer (which we call “clear” for short) of $1$ mm, whose lengths are
approximately 70% the length of the face. A second type of sample prepared in
a similar fashion, but with a centered cylindrical hole of 3 mm diameter
drilled in it parallel to the surfaces of the sample in all cases mentioned,
was also used in the experiments in order to compare the signals with the
former. This cavity was alternatively emptied or filled with deionized water.
We call “sample 1” the one drilled with the cylindrical cavity, and “sample 2”
the one without the hole. Figure 2 is a sketch of sample 1 with a schematic
representation of the fiduciary pattern used. The PZT and the laser beam
relative positions are displayed, together with the approximate position of
the cavity.

Figure 2: The Grilon®sample prepared for the surface absorption experiments.
The shadowed region (PZT) is the location of the transducer with respect to
the impinging laser beam direction (LBD). The cavity is a $3$ mm diameter
hole, whose position (HP) is shown for the samples that have drilled cavities.
The cavity may be empty or filled with water. The height of the samples is
$39$ mm and has a $10$ mm square base. It has five grooves (FP) in its front
face, painted in black to enhance absorption.

We obtained two types of signals, those from samples without holes and those
from samples with centered holes. Each type was subdivided into signals taken
with the laser impinging on the blank surface, and those taken with the laser
impinging on the patterned surface. Besides, there are signals obtained from
the samples with cavities, either empty or filled with water.

In each sample, the laser point of impact was moved from the farthest possible
position to the nearest with respect to the PZT. This was accomplished by
means of a $1\,\mu$m precision, step motor movable stage, Zaber Model T-LA60A,
controlled by a PC interface.

##  3 Results

In order to properly analyze the results, we calibrate the response of the
system to increasing laser pulse energy. To this end we irradiate a blank
surface of sample 2 at a point near the center of the face, and we plotted the
amplitude of the first maximum of the acoustic signal as a function of the
laser pulse energy. The result is displayed in Fig. 3 and shows linearity in
the energy range used.

Figure 3: PAS vs. laser pulse energy. The linearity of the PA response
(squares) to laser excitation is evident in the plots. The black dots
represent the PA signal when the laser impinges on a black stripe nearly at
the center of the front face of a striped sample.

In the same plot, we display three points (including the origin) which are the
maxima of the signal at the same location of a striped sample face, but
impinging on a black groove. As it can be seen, the signal nearly trebles its
maximum peak for the same excitation energy. In both experiments, the pinhole
used was $1.5$ mm in diameter.

After ascertaining the linearity of the response, and the fact that there is
an evident dependence of the PAS on the absorbance of the surface, we obtain a
profile of three of the grooves of sample 2 by plotting the value of the
amplitude of the first peak of the PA signal versus the relative distance
between the PZT and the excited region using the same pinhole as before. The
result of this is shown in Fig. 4. It can be seen that 1) the groove profile
is neatly resolved, and 2) there is an improvement of the signal generated in
the black stripes which decreases as the distance increases. Since the stripes
and the laser beam have approximately the same transverse size of the grooves,
the resulting profile is somehow rounded off, but this is not important in
what we aim to prove here.

Figure 4: The centers of clear bands and black grooves are clearly resolved by
scanning the surface with the laser. The signals were taken at $100\,\mu$m
displacement from each other. The energy of PAS decreases with distance of the
laser beam to PZT. The increment in signal due to the grooves is more than
6-fold with respect to the signal due to the bulk polymer.

We could determine the speed of sound in the sample from a plot of the time
position of the beginning of the first peak of the acoustic signal (arrival
time), as a function of the distance between the impinging point on the sample
and the PZT detector. But when we try to do that, experiments demonstrate that
the time elapsed since a laser pulse triggers a digital oscilloscope and the
appearance of the PA signal is the same regardless of the distance between the
impinging laser beam and the detector, due mainly to the light scattered by
the bulk of the polymer that hits the PZT. This poses a problem in the
evaluation of the speed of the waves.

To avoid that difficulty, we use the signal produced if the laser hits in the
black grooves, generated only by the absorption at the grooves. We obtain this
by subtracting from the PAS signal measured when the laser impinges in a black
groove, the PAS signal obtained in a “clear” region nearby. To this end, we
moved the sample slightly away from the previous black stripe, so the first
PAS signal was obtained with the beam impinging in a black groove and the
second PAS signal was obtained with the beam impinging in a white stripe.

Figure 5 shows the determination of the speed of sound in sample 2 by this
method. Since the plot uses as input the maxima of the amplitude of the
signals, the straight line would not cross the origin. The extrapolated value
for zero-crossing corresponds approximately to the amplitude of the first
maximum of the signal in clear samples.

Figure 5: Time of the first maximum of the processed signals vs. the position
of the impinging point of the laser presents a linear correlation
($R\,\approx\,0.994$) that yields a value of $v=(2333\,\pm\,133)$ ms-1 for the
speed of sound. All these signals were taken with a pinhole of $1$ mm
diameter.

Evaluation of the slope of the resulting line allows the calculation of the
value of speed, as $v=(2333\pm 133)$ ms-1, which compares well with calculated
data determined by using the properties of the polymer [22]. Since the
vibration of the whole sample has distinctive frequencies, the FFT of the PA
signal provides another estimation of the speed of waves, once the frequency
sequence is properly found. The Fast Fourier Transform (FFT) analysis was used
both, to estimate the sound speed in the Grilon®sample via the frequencies
identified in the spectrum and their spacing, knowing that the piece is a
parallelepiped of known dimensions, and to define a scale for the energy of
the pulse. The details of this procedure are straightforward calculations
[23]. We found it useful to use the power spectrum for defining the energy
instead of the integral of the temporal pulse and that is the parameter we use
in the presentation of the results.

Figures 6 and 7 show an FFT treatment of the signals obtained from the
following three cases: solid Grilon®sample, sample with empty cavity and
sample with the hole filled with deionized water. The PAS energy used in this
figure is a measure of the energy content of the acoustic pulse, as evaluated
from the power spectrum of the signal.

In Fig. 6, we display the results obtained impinging with the laser on clear
faces, and in Fig. 7 we show the results impinging with the laser on the
patterned faces. It is clear that a distinctive feature arises near the center
of the sample in the patterned faces where a black stripe is located, which is
not visible in the clear-face analysis.

Figure 6: The acoustic energy (FFT power integral of the PAS) vs. the relative
distance between the laser beam and the PZT in Grilon®samples in a face with
no fiduciary pattern (clear sample). The energy diminishes as the distance to
the detector increases. References in the insert: Triangles represent the
cavity empty in a clear sample. Circles are for cavity filled with water in
clear samples. Rhombi are for clear samples without the cavity. Figure 7: The
energy deposited by the laser on a white stripe and on the black grooves for
the three cases analyzed as seen in the insert. The circles represent a
grooved sample with the cavity filled with water. Triangles and rhombi are for
grooved samples and the empty cavity, and no cavity, respectively.

The acoustic energy deposited in the patterned faces is more than one order of
magnitude higher than that obtained in the clear faces when the inclusion is
present (compare Fig. 6 with Fig. 7). The water-filled hole and the empty hole
are also clearly distinguishable from the solid Grilon®response.

##  4 Analysis and Conclusions

We have performed an experimental analysis of the photoacustic signal in
Grilon®polymer but it can be extended to other materials as epoxy resins used
also as biological phantoms. We have shown that such applications are viable
for quantitative determinations. The above results can be used to determine
the presence and some optical characteristics of an inhomogeneity embedded in
this type of materials.

All the acoustic signals detected by the PZT in this soft turbid solid
material begin at approximately the same time after the laser trigger fires,
regardless of the relative distance from the impact point of the excitation to
the PZT acoustic detector, due mainly to scattering, making this method
useless to evaluate the speed of sound by the scanning standard procedure.

The differences in those PA signals are difficult to analyze. Therefore, to
evaluate the speed of the acoustic waves and to gather information about the
presence of a cavity or inhomogeneity in the polymer, we have developed a
method that used a regular pattern on the surface of the sample, consisting on
parallel clear stripes of the base material and grooves filled with highly
absorbent black paint. In the black grooves, the localized absorption provides
a strong shock wave at the surface. By comparing the time of appearance of the
signal in different positions of the surface, it is possible to estimate the
speed of those waves in the polymer.

For each of the zones in the pattern, the PA curves undergo a change of shape
and amplitude from signals in the white zones to the signals obtained in the
black stripes, being this strong evidence of the effect of the inhomogeneity
in the signal. A Power FFT was performed on each signal in order to provide
another means to determine whether a black or a white stripe is excited, and
the integral of the FFT provides a measure of the energy absorbed by the
sample in each case. Please note that the energy of the laser was fixed to a
value that avoids bleaching of the paint, being in all cases below $500\,\mu$J
per pulse.

The signals generated at these inhomogeneities provide a well defined point of
absorption and thus a definite path for the sound generated by the light-
absorption mechanism which is very distinctive from other mechanisms of
excitation of the PZT. It also reveals the presence of a surface inhomogeneity
once the contributions of other sources of acoustic waves are identified
making suitable use of reference signals. All the conclusions of this work
must be under the proviso that the PZT has a limited frequency band.

Although the above results were obtained for a soft polymer with a fiduciary
painted pattern, they can be extended to other type of resins with charge of
dyes or other absorbent particles. We are confident that with minor
modifications it can be used for the determination of properties of materials
of biological interest as well.

The results shown in Figs. 6 and 7 confirm that employing one of the surfaces
of the sample conveniently patterned, and scanning it for detection of
ultrasound signals, can be used to determine the presence of an inhomogeneity,
albeit its precise location and size is not well defined by this procedure and
it should be complemented by similar determinations at other relative
positions of the laser and the PZT. The increase in the signal with respect to
the background material is at least one order of magnitude or better.

When using this technique in phantoms used in medical applications, one should
take care of the fact that there are limitations in several aspects, such as
the power involved in each pulse avoiding any kind of damage, and that using
other wavelengths would be better suited for biological tissues which involve
blood. Other type of samples are being currently inspected by modifications of
the procedure reported here so to adapt it to gelled phantoms.

The conclusions are, in short, that the system is sensitive to the presence of
the inhomogeneity, and that the higher absorbance of the painted stripes in
the surface allows not only to evaluate the speed of sound (which is essential
to any tomographic technique) but also improves the detectivity by enhancing
the energy released as mechanical waves. This is a non-trivial result since in
the modeling of the propagation of the laser light in the turbid substance,
scattering is predominant, but still is sufficient for the detection of
inhomogeneities through changes in the absorption. The technique based on the
PA is simple and has the advantage that it can be adapted to be used in larger
samples or in samples of biological interest. The procedure of using a single
acoustic detector for the signals produced by the laser scanning of the
surface under study, has an advantage over multiple detector arrangements in
the sense that with a suitable fiduciary pattern the method can provide
information about the speed of the waves involved in the signal. This is
interesting because the data processing would not depend on generic
information about its value.

###### Acknowledgements.

PG wants to thank the Red Nacional de Laboratorios de Óptica for financial
help and partial funding during the experiments and to InterU System for
providing a grant for the completion of the experiments. This work partially
funded by Universidad Nacional del Centro de la Provincia de Buenos Aires,
Agencia Nacional de Promoción Científica y Tecnológica (PICT 0570) and CONICET
(PIP 384). HODR, DII, JAP and HFRS are members of Carrera del Investigador
Científico, Consejo Nacional de Investigaciones Científicas y Técnicas
(Argentina). GMB is member of Carrera del Investigador Científico, Comisión de
Investigaciones Científicas de la Provincia de Buenos Aires (Argentina).
Authors wish to thank Nicolás A. Carbone for help in the final preparation of
the manuscript.

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# On Gromov’s conjecture for right-angled Artin groups

Alexander Dranishnikov1 and  Satyanath Howladar  Alexander N. Dranishnikov,
Department of Mathematics, University of Florida, 358 Little Hall,
Gainesville, FL 32611-8105, USA<EMAIL_ADDRESS>Satyanath Howladar,
Department of Mathematics, University of Florida, 358 Little Hall,
Gainesville, FL 32611-8105, USA<EMAIL_ADDRESS>
###### Abstract.

We explore in terms of KO-homology the 2nd obstruction for the universal
covering $\widetilde{M}$ of a closed spin $n$-manifold $M$ with positive
scalar curvature (PSC) to have macroscopic dimension $\leq n-2$. As a
corollary we prove that the universal covering $\widetilde{M}$ of an
$n$-dimensional closed spin PSC manifold $M$ with $\pi_{1}(M)\in$ RAAG has
macroscopic dimension $\dim_{mc}\widetilde{M}\leq n-2$. This confirms Gromov’s
conjecture in the case of RAAG (right-angled Artin group) fundamental groups.

###### Key words and phrases:

###### Key words and phrases:

positive scalar curvature, inessential manifold, macroscopic dimension, right-
angled Artin groups

###### 2000 Mathematics Subject Classification:

Primary 53C23, Secondary 57N65, 19L41, 19M05, 20F36

1Supported by Simons Foundation

##  1\. Introduction

The notion of macroscopic dimension was introduced by M. Gromov [G2] to study
topology of manifolds with a positive scalar curvature metric.

######  1.1 Definition.

A metric space $X$ has the macroscopic dimension $\dim_{\text{mc}}X\leq k$ if
there is a uniformly cobounded proper map $f:X\to K$ to a $k$-dimensional
simplicial complex. Then $\dim_{mc}X=m$ where $m$ is minimal among $k$ with
$\dim_{\text{mc}}X\leq k$.

A map of a metric space $f:X\to Y$ is uniformly cobounded if there is a
uniform upper bound on the diameter of preimages $f^{-1}(y)$, $y\in Y$.

Gromov’s Conjecture. The macroscopic dimension of the universal covering
$\widetilde{M}$ of a closed positive scalar curvature $n$-manifold $M$
satisfies the inequality $\dim_{mc}\widetilde{M}\leq n-2$ for the metric on
$\widetilde{M}$ lifted from $M$.

The main examples supporting Gromov’s Conjecture are $n$-manifolds of the form
$M=N\times S^{2}$. They admit metrics with PSC in view of the formula
$Sc_{x_{1},x_{2}}=Sc_{x_{1}}+Sc_{x_{2}}$ for the cartesian product
$(X_{1}\times X_{2},\mathcal{G}_{1}\oplus\mathcal{G}_{2})$ of two Riemannian
manifolds $(X_{1},\mathcal{G}_{1})$ and $(X_{2},\mathcal{G}_{2})$ and the fact
that while $Sc_{N}$ is bounded $Sc_{S^{2}}$ can be chosen to be arbitrary
large. Clearly, the projection $p:\widetilde{M}=\widetilde{N}\times
S^{2}\to\widetilde{N}$ is a proper uniformly cobounded map to a
$(n-2)$-dimensional manifold which can be triangulated. Hence,
$\dim_{mc}\widetilde{M}\leq n-2$.

Since $\dim_{mc}X=0$ for every compact metric space, the Gromov Conjecture
holds trivially for manifolds with finite fundamental groups. Therefore,
Gromov’s conjecture is about manifolds with infinite fundamental groups. We
note that even a weaker version of the Gromov Conjecture, that is
$\dim_{mc}\widetilde{M}\leq n-1$, for positive scalar curvature manifolds
looks out of reach. This is because it implies the Gromov-Lawson’s Conjecture:
A closed aspherical manifold cannot carry a metric of positive scalar
curvature. The latter is known as a twin sister of the famous Novikov Higher
Signature conjecture. The real analytic version of the Novikov conjecture is
called the Strong Novikov Conjecture. It is a conjecture for discrete groups
$\pi$: The analytic assembly map $\alpha:KO_{*}(B\pi)\to KO_{*}(C^{*}(\pi))$
is a monomorphism, where $C^{*}(\pi)$ is reduced $C^{*}$-algebra of $\pi$. We
note that the Strong Novikov conjecture for a group $\pi$ implies the Higher
Signature conjecture for all manifolds with the fundamental group $\pi$.

The (Strong) Novikov’s conjecture is proven for large classes of groups (see
the survey [Yu]). The Gromov Conjecture is proven in far fewer cases
[Bo1],[Bo3], [Dr1],[Dr2], [BD1],[BD2],[DD]. In view of the above connection
between the Gromov and the Novikov conjectures, it makes sense to state the
Gromov Conjecture for groups instead of manifolds as the Gromov Conjecture for
all positive scalar curvature manifolds with a given fundamental group $\pi$.
Additionally, to make Gromov’s conjecture more accessible we consider it in
this paper for spin manifolds with the fundamental group $\pi$ satisfying the
Strong Novikov Conjecture. In this paper we proved the inequality
$\dim_{mc}\widetilde{M}\leq n-1$ for spin manifolds with the fundamental group
$\Gamma$ that satisfies the Strong Novikov Conjecture and has its integral
homology $H_{*}(\Gamma)$ torsion free (Corollary 2.11).

Gromov defined inessential manifolds $M$ as those for which a classifying map
$u:M\to B\Gamma$ of the universal covering $\widetilde{M}$ can be deformed to
the $(n-1)$-skeleton $B\Gamma^{(n-1)}$ where $n=\dim M$. Clearly, for an
inessential $n$-manifold $M$ we have $\dim_{mc}\widetilde{M}\leq n-1$.

We call an $n$-manifold strongly inessential if a classifying map of its
universal covering $u:M\to B\Gamma$ can be deformed to the $(n-2)$-skeleton.
Since for strongly inessential $n$-manifolds $\dim_{mc}\widetilde{M}\leq n-2$,
the following conjecture implies the Gromov’s conjecture.

Strong Gromov’s Conjecture. A closed positive scalar curvature manifold $M$ is
strongly inessential.

We note that for a finite cover $M^{\prime}\to M$ of a PSC manifold $M$ the
manifold $M^{\prime}$ has PSC and the universal cover of $M^{\prime}$
coincides with the universal cover of $M$, Therefore, the Strong Gromov
Conjecture for a group $\Gamma^{\prime}$ implies the original Gromov
Conjecture for all groups $\Gamma$ containing $\Gamma^{\prime}$ as a finite
index subgroup.

Dmitry Bolotov showed [Bo2] that there are inessential manifolds which are not
strongly inessential.

In this paper we prove the Strong Gromov Conjecture for spin $n$-manifolds
with $n>4$, for some classes of groups, in particular, for right-angled Artin
groups (RAAGs).

##  2\. Preliminaries

We recall that for every spectrum $E$ there is a connective cover $e\to E$,
that is a spectrum $e$ with the morphism $e\to E$ that induces isomorphisms of
homotopy groups $\pi_{i}(e)\to\pi_{i}(E)$ for $i\geq 0$ and with
$\pi_{i}(e)=0$ for $i<0$ . By $KO$ we denote the spectrum for real K-theory,
by $ko$ its connective cover, and by $per:ko\to KO$ the corresponding morphism
of spectra. For more details we refer to [Ru]. We use the standard notation
$\pi^{s}_{*}$ for the stable homotopy groups. We use both notations for an
$E$-homology of a space $X$: old-fashioned $E_{*}(X)$ and modern $H_{*}(X;E)$.
We recall that $KO_{n}(pt)={\mathbb{Z}}$ if $n=0$ or $n=4$ mod 8,
$KO_{n}(pt)={\mathbb{Z}}_{2}$ if $n=1$ or $n=2$ mod 8, and $KO_{n}(pt)=0$ for
all other values of $n$. By $\mathbb{S}$ we denote the spherical spectrum.
Note that for any ring spectrum $E$ there is a natural morphism $\mathbb{S}\to
E$ which leads to the natural transformation of the stable homotopy to
$E$-homology $\pi^{s}_{*}(X)\to H_{*}(X;E)$.

Let $M({\mathbb{Z}}_{m},n)=S^{n}\cup_{\phi}D^{n+1}$ denote the Moore space
where $\phi:S^{n}\to S^{n}$ is a map of degree $m$.

######  2.1 Proposition.

The homomorphism

$per:ko_{i}(M({\mathbb{Z}}_{m},n))\to KO_{i}(M({\mathbb{Z}}_{m},n))$

is an isomorphism for $i\geq n$.

###### Proof.

We apply the Five Lemma to the diagram with $M=M({\mathbb{Z}}_{m},n)$

$\setcounter{MaxMatrixCols}{11}\begin{CD}ko_{i}(S^{n})@>{m}>{}>ko_{i}(S^{n})@>{}>{}>ko_{i}(M)@>{}>{}>ko_{i-1}(S^{n})@>{m}>{}>ko_{i-1}(S^{n})\\\
@V{\cong}V{}V@V{\cong}V{}V@V{}V{}V@V{\cong}V{}V@V{\cong}V{}V\\\
KO_{i}(S^{n})@>{m}>{}>KO_{i}(S^{n})@>{}>{}>KO_{i}(M)@>{}>{}>KO_{i-1}(S^{n})@>{m}>{}>KO_{i-1}(S^{n})\\\
\end{CD}$

to get the result. Note that for $i=n$ on the right there will be
isomorphisms, since $KO_{-1}(pt)=0$. ∎

The following proposition is taken from [BD1].

######  2.2 Proposition.

The natural transformation $\pi_{*}^{s}(pt)\to ko_{*}(pt)$ induces an
isomorphism $\pi_{n}^{s}(K/K^{(n-2)})\to ko_{n}(K/K^{(n-2)})$ for any CW
complex $K$.

We use the following well-known fact (see 4C, [Ha]).

######  2.3 Proposition.

Let $X$ be an $(n-1)$-connected $(n+1)$-dimensional CW complex. Then $X$ is
homotopy equivalent to the wedge of spheres of dimensions $n$ and $n+1$
together with the Moore spaces $M({\mathbb{Z}}_{m},n)$.

Next, we recall the following classical result (see Corollary 6.10.3, [TtD]):

######  2.4 Theorem.

Suppose that a CW-complex pair $(X,A)$ satisfies the conditions $\pi_{i}(X)=0$
for $i\leq m$ and $\pi_{i}(A)=0$ for $i\leq m-1$ with $m\geq 2$. Then the
quotient map $q:(X,A)\to(X/A,\ast)$ induces isomorphisms
$q_{*}:\pi_{i}(X,A)\to\pi_{i}(X/A)$ for $i\leq 2m-1$.

###  2.1. Inessential manifolds

Note that for an inessential $n$-manifold $M$ we have
$\dim_{mc}\widetilde{M}\leq n-1$. Indeed, a lift
$\widetilde{u_{M}}:\widetilde{M}\to E\Gamma^{(n-1)}$ of a classifying map is a
uniformly cobounded proper map to an $(n-1)$-complex.

The first step in any proof of the Strong Gromov Conjecture would be
establishing the inessentiality of a positive scalar curvature manifold. We
recall that the inessentiality of a manifold can be characterized as follows
[Ba] (see also [BD1], Proposition 3.2).

######  2.5 Theorem.

Let $M$ be a closed oriented $n$-manifold. Then the following are equivalent:

1\. $M$ is inessential;

2\. $(u_{M})_{*}([M])=0$ in $H_{n}(B\Gamma)$ where $[M]$ is the fundamental
class of $M$ and $\Gamma=\pi_{1}(M)$.

Gromov calls an orientable closed $n$-manifold $M$ rationally inessential if
$(u_{M})_{*}([M])=0$ in $H_{n}(B\Gamma;{\mathbb{Q}})$.

It is known that for a manifold to be spin is equivalent to be orientable for
any of the K-theories: complex $KU$, real $KO$, or their connective covers
$ku$ or $ko$ [Ru]. Suppose that a closed $n$-manifold $M$ is orientable with
respect to a generalized homology theory $E$. We call the manifold $M$ to be
$E$-inessential if it is $E$-orientable and $u_{*}([M]_{E})=0$ where
$[M]_{E}\in H_{n}(M;E)$ is the $E$-fundamental class. Note that this property
does not depend on the choice of $E$-orientation.

We recall the Rosenberg index theorem [R1]:

######  2.6 Theorem.

Suppose that the fundamental group $\Gamma$ of a positive scalar curvature
spin manifold $M$ satisfies the Strong Novikov Conjecture. Then $M$ is
$KO$-inessential.

Note that in view of the Chern character isomorphism a $KO$-inessential
manifold is rationally inessential. As Marcinkowski’s example shows [M], the
rational inessentiality of an $n$-manifold does not imply the inequality
$\dim_{mc}\widetilde{M}\leq n-1$. This makes difficult the first step in the
Gromov’s conjecture even in the case of groups satisfying the Strong Novikov
Conjecture.

######  2.7 Proposition ( [BD1], Proposition 3.3).

A $ko$-inessential manifold is inessential.

######  2.8 Question.

When a $KO$-inessential manifold is inessential ?

In view of Proposition 2.7 this is true for manifolds with the fundamental
group $\Gamma$ for which the map $per:ko_{n}(B\Gamma)\to KO_{n}(B\Gamma)$ is
injective.

In view of the Rosenberg Index theorem, we obtain the following

######  2.9 Proposition.

Suppose that the fundamental group $\Gamma$ of a positive scalar curvature
closed spin $n$-manifold $M$ has the following properties:

(1) It satisfies the Strong Novikov Conjecture;

(2) The map $per:ko_{n}(B\Gamma)\to KO_{n}(B\Gamma)$ is injective.

Then $M$ is inessential.

The conditions (1) and (2 for all $n$) are known as the Rosenberg-Stolz
conditions. For closed spin manifolds with the fundamental group satisfying
Rosenberg-Stoltz conditions there is a vanishing index criterion when they
could carry a metric with positive scalar curvature [R2]. In other words the
Gromov-Lawson-Rosenberg conjecture holds true for such groups.

######  2.10 Proposition.

Suppose that a group $\Gamma$ has torsion free integral homology. Then every
$KO$-inessential manifold with the fundamental group $\Gamma$ is inessential.

###### Proof.

Since the $KO$-fundamental class $[M^{n}]_{KO}$ maps under the natural
transformation $KO\to KU$ to a $KU$-fundamental class, every $KO$-inessential
manifold is $KU$-inessential. In view of the diagram of the homology Chern
character isomorphisms generated by a degree one map $f:M^{n}\to S^{n}$

$\begin{CD}K_{n}(M^{n})\otimes{\mathbb{Q}}@>{ch}>{\cong}>\bigoplus_{i\in{\mathbb{Z}}}H_{n+2i}(M^{n})\otimes{\mathbb{Q}}\\\
@V{f_{*}\otimes 1_{{\mathbb{Q}}}}V{}V@ Vf_{*}\otimes 1_{{\mathbb{Q}}}VV\\\
K_{n}(S^{n})\otimes
1_{{\mathbb{Q}}}@>{ch}>{\cong}>\bigoplus_{i\in{\mathbb{Z}}}H_{n+2i}(S^{n})\otimes{\mathbb{Q}}\\\
\end{CD}$

the image of a $KU$-fundamental class of $M^{n}$ under the Chern character has
nontrivial $H_{n}(M^{n})$ component. Then the bottom arrow in the diagram

$\begin{CD}H_{n}(M^{n})@>{u_{*}}>{}>H_{n}(B\Gamma)\\\ @V{}V{}V@V{}V{}V\\\
H_{n}(M^{n})\otimes{\mathbb{Q}}@>{u_{*}\otimes
1_{{\mathbb{Q}}}}>{}>H_{n}(B\Gamma)\otimes{\mathbb{Q}}\\\ \end{CD}$

is a zero homomorphism. The right vertical homomorphism is injective, since
$H_{n}(B\Gamma)$ is torsion free. Therefore, the top arrow is zero. ∎

######  2.11 Corollary.

Suppose that the fundamental group $\Gamma$ of a positive scalar curvature
spin manifold $M$ satisfies the Strong Novikov conjecture and all homology
groups $H_{*}(\Gamma)$ are torsion free. Then

$\dim_{mc}\widetilde{M}\leq n-1.$

###### Proof.

We apply Rosenberg’s theorem (Theorem 2.6) and Proposition 2.10. ∎

In [BD1] we proved the following addendum to Theorem 2.5.

######  2.12 Proposition ([BD1], Lemma 3.5).

For an inessential manifold $M$ with a CW complex structure a classifying map
$u:M\to B\Gamma$ can be chosen such that

$u(M,M^{(n-1)})\subset(B\Gamma^{(n-1)},B\Gamma^{(n-2)}).$

##  3\.  The second obstruction

Suppose that $\Gamma$ acts on an abelian group $\pi$. By
$\pi_{\Gamma}=\pi\otimes_{{\mathbb{Z}}\Gamma}{\mathbb{Z}}$ we denote the group
of coinvariants. Since every $ko$-inessential manifold is inessential (
Proposition 2.7) and for every inessential $n$-manifold there is a classifying
map $u:M\to B\Gamma$ with
$u(M,M^{(n-1)})\subset(B\Gamma^{(n-1)},B\Gamma^{(n-2)})$ ( Proposition 2.12),
for $ko$-inessential manifolds we always will consider such classifying maps
$u$.

The following lemma shows that the second step in the Strong Gromov’s
conjecture can be also reduced to a question about $ko$-inessentiality.

######  3.1 Lemma.

Suppose that a closed spin $n$-manifold $M$, $n>4$, with $\Gamma=\pi_{1}(M)$
is $ko$-inessential in a way that $u_{*}([M]_{ko})=0$ in
$ko_{n}(B\Gamma^{(n)})$ for a cellular classifying map $u:M\to B\Gamma$. Then
$M$ is strongly inessential.

Moreover, $M$ is strongly inessential if $j_{*}u_{*}([M]_{ko})=0$ where
$j:B\Gamma^{(n)}\to B\Gamma^{(n)}/B\Gamma^{(n-2)}$ is the quotient map.

###### Proof.

We may assume that $M$ has a CW complex structure with one $n$-dimensional
cell. Let $\psi:D^{n}\to M$ be its characteristic map.

We show that the lifting problem

$\textstyle{M^{(n-1)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\textstyle{B\Gamma^{(n-2)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\subset}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{iu}$$\textstyle{B\Gamma^{(n)}.}$

has a solution. Here $i:B\Gamma^{(n-1)}\to B\Gamma^{(n)}$ is the inclusion. It
would mean that there is a homotopy lift $\hat{u}:M\to B\Gamma^{(n-2)}$ of
$i\circ u$ which agrees with $u$ on $M^{(n-2)}$. Since $n\geq 4$, the map
$\hat{u}$ induces an isomorphism of the fundamental groups and, hence, is
classifying map.

Note that the $(n-1)$-homotopy groups of the homotopy fiber $F$ of the
inclusion $B\Gamma^{(n-2)}\to B\Gamma^{(n)}$ equals the relative $n$-homotopy
group, $\pi_{n-1}(F)=\pi_{n}(B\Gamma^{(n)},B\Gamma^{(n-2)})$. Then the first
and the only obstruction to deform $u$ to $B\Gamma^{(n-2)}$ is defined by the
cocycle $c_{u}:C_{n}(M)\to\pi_{n-1}(F)$ represented by the composition

$\pi_{n}(D^{n},\partial
D^{n})\stackrel{{\scriptstyle\psi_{*}}}{{\to}}\pi_{n}(M,M^{(n-1)})\stackrel{{\scriptstyle
u_{*}}}{{\to}}\pi_{n}(B\Gamma^{(n-1)},B\Gamma^{(n-2)})\stackrel{{\scriptstyle
i_{*}}}{{\to}}\pi_{n}(B\Gamma^{(n)},B\Gamma^{(n-2)})$

with the cohomology class $o_{u}=[c_{u}]\in
H^{n}(M;\pi_{n}(B\Gamma^{(n)},B\Gamma^{(n-2)}))$. By the Poincare Duality with
local coefficients, the cohomology class $o_{u}$ is dual to the homology class

$PD(o_{u})\in
H_{0}(M;\pi_{n}(B\Gamma^{(n)},B\Gamma^{(n-2)}))=\pi_{n}(B\Gamma^{(n)},B\Gamma^{(n-2)})_{\Gamma}$

represented by $q_{*}i_{*}u_{*}\psi_{*}(1)$ where

$q_{*}:\pi_{n}(B\Gamma^{(n)},B\Gamma^{(n-2)})\to\pi_{n}(B\Gamma^{(n)},B\Gamma^{(n-2)})_{\Gamma}$

is the projection onto the group of coinvariants.

Note that
$\pi_{n}(B\Gamma^{(n-1)},B\Gamma^{(n-2)})=\pi_{n}(E\Gamma^{(n-1)},E\Gamma^{(n-2)})$.
Below we will identify these groups. Denote by

$\tilde{i}:(E\Gamma^{(n-1)},E\Gamma^{(n-2)})\to(E\Gamma^{(n)},E\Gamma^{(n-2)})$

the inclusion induced by $i$.

Since $n\leq 2(n-2)-1$, by Theorem 2.4,

$\pi_{n}(E\Gamma^{(n-1)},E\Gamma^{(n-2)})=\pi_{n}(E\Gamma^{(n-1)}/E\Gamma^{(n-2)}).$

It is easy to see that

$\pi_{n}(E\Gamma^{(n-1)}/E\Gamma^{(n-2)})_{\Gamma}=\pi_{n}(B\Gamma^{(n-1)}/B\Gamma^{(n-2)}).$

Similarly,

$\pi_{n}(E\Gamma^{(n)}/E\Gamma^{(n-1)})_{\Gamma}=\pi_{n}(B\Gamma^{(n)}/B\Gamma^{(n-1)}).$

The homotopy exact sequence of the triple
$(E\Gamma^{(n)},E\Gamma^{(n-1)},E\Gamma^{(n-2)})$ produces the following
commutative diagram

$\begin{CD}\pi_{n+1}(E\Gamma^{(n)},E\Gamma^{(n-1)})@>{}>{}>\pi_{n}(E\Gamma^{(n-1)},E\Gamma^{(n-2)})@>{\tilde{i}_{*}}>{}>im(\tilde{i}_{*})@>{}>{}>0\\\
@V{}V{}V@V{}V{}V@V{q_{*}}V{}V\\\
\pi_{n+1}(E\Gamma^{(n)},E\Gamma^{(n-1)})_{\Gamma}@>{}>{}>\pi_{n}(E\Gamma^{(n-1)},E\Gamma^{(n-2)})_{\Gamma}@>{\tilde{i}_{*}\otimes_{\Gamma}1_{\mathbb{Z}}}>{}>im(\tilde{i}_{*})_{\Gamma}@>{}>{}>0\\\
@V{\cong}V{}V@V{\cong}V{}V@V{\xi}V{}V\\\
\pi_{n+1}(B\Gamma^{(n)}/B\Gamma^{(n-1)})@>{}>{}>\pi_{n}(B\Gamma^{(n-1)}/B\Gamma^{(n-2)})@>{i_{*}}>{}>im(i_{*})@>{}>{}>0\\\
\end{CD}$

where the middle row is exact being obtained by tensor product by
${\mathbb{Z}}$ over ${\mathbb{Z}}\Gamma$ of an exact row. By the Five Lemma
the homomorphism $\xi$ is an isomorphism.

Denote by $\bar{u}:M/M^{(n-1)}=S^{n}\to B\Gamma^{(n-1)}/B\Gamma^{(n-2)}$ the
induced map. The following commutative diagram

$\begin{CD}\pi_{n}(M,M^{(n-1)})@>{\tilde{i}_{*}u_{*}}>{}>im(\tilde{i}_{*})@>{q_{*}}>{}>im(\tilde{i}_{*})_{\Gamma}\\\
@A{\psi_{*}}A{}A@V{}V{\xi}V\\\ \pi_{n}(D^{n},\partial
D^{n})@>{=}>{}>\pi_{n}(M/M^{(n-1)})@>{i_{*}\bar{u}_{*}}>{}>im(i_{*})@>{\subset}>{}>\pi_{n}(B\Gamma^{(n)}/B\Gamma^{(n-2)})\\\
\end{CD}$

implies that $i_{*}\bar{u}_{*}(1)=\xi q_{*}\tilde{i}_{*}u_{*}\psi_{*}(1).$
Thus, $i_{*}\bar{u}_{*}(1)=0$ if and only if the obstruction $o_{u}$ vanishes.

We show that $i_{*}\bar{u}_{*}(1)=0$. The restriction $n>4$ imply that
$\bar{u}_{*}(1)$ survives to the $ko$-homology group:

$\begin{CD}\pi_{n}(B\Gamma^{(n)}/B\Gamma^{(n-2)})@>{\cong}>{}>\pi_{n}^{s}(B\Gamma^{(n)}/B\Gamma^{(n-2)})@>{\cong}>{}>ko_{n}(B\Gamma^{(n)}/B\Gamma^{(n-2)}).\\\
\end{CD}$

The first isomorphism is obtained by taking $n$-times suspension and using
Freudenthal Theorem, and the second isomorphism by Proposition 2.2. Then the
commutative diagram

$\begin{CD}\pi_{n}(S^{n})@>{\cong}>{}>ko_{n}(S^{n})\\\
@V{i_{*}\bar{u}_{*}}V{}V@V{i_{*}\bar{u}_{*}}V{}V\\\
\pi_{n}(B\Gamma^{(n)}/B\Gamma^{(n-2)})@>{\cong}>{}>ko_{n}(B\Gamma^{(n)}/B\Gamma^{(n-2)})\end{CD}$

implies that $i_{*}\bar{u}_{*}(1)=0$ for $ko_{n}$ if and only if
$i_{*}\bar{u}_{*}(1)=0$ for $\pi_{n}$.

From the assumption and the diagram defined by the quotient maps
$j^{\prime}:M\to M/M^{(n-1)}=S^{n}$ and $j:B\Gamma^{(n)}\to
B\Gamma^{(n)}/B\Gamma^{(n-2)}$

$\begin{CD}ko_{n}(M)@>{u_{*}}>{}>ko_{n}(B\Gamma^{(n)})\\\
@V{j_{*}^{\prime}}V{}V@V{j_{*}}V{}V\\\
ko_{n}(S^{n})@>{i_{*}\bar{u}_{*}}>{}>ko_{n}(B\Gamma^{(n)}/B\Gamma^{(n-2)})\\\
\end{CD}$

it follows that
$i_{*}\bar{u}_{*}(1)=i_{*}\bar{u}_{*}j^{\prime}_{*}([M]_{ko})=j_{*}u_{*}([M]_{ko})=0$.
∎

######  3.2 Remark.

A slightly weaker lemma was proven in [Dr1]. Namely, a strong inessentiality
of $M$ was proven under assumption of $j_{*}u_{*}([M]_{ko})=0$ in
$ko_{n}(B\Gamma^{(n-1)}/B\Gamma^{(n-2)})$ instead of
$ko_{n}(B\Gamma^{(n)}/B\Gamma^{(n-2)})$. Since $ko_{*}$ is a connective
homology theory, for any $ko$-inessential $n$-manifold $M$ with the
fundamental group $\Gamma$ we have $u_{*}([M]_{ko})=0$ in
$ko_{n}(B\Gamma^{(n+1)}/B\Gamma^{(n-2)})$. Thus, in the current version of
lemma on the second obstruction we are one step closer to an affirmative
answer to the following question.

######  3.3 Question.

Is every $ko$-inessential manifold strongly inessential ?

We note that Bolotov’s example [Bo2] is $ko$-essential.

##  4\. K-theory Stabilization Condition

The following condition on K-theory of a group $\Gamma$ was introduced in
[Dr1] In the proof of the Strong Gromov Conjecture for abelian fundamental
groups in [Dr1] (see also [Bo1]) we considered the following property of
groups $\Gamma$ denoted by (*) : The inclusion homomorphism
$KO_{*}(B\Gamma^{(m)})\to KO_{*}(B\Gamma)$ is injective for all $m$.

This property depends on choice of $B\Gamma$. Here we introduce a weaker
condition which is independent of the choice of $B\Gamma$ and sufficient for
our purposes.

First, we define this condition for spaces. We state it for arbitrary
generalized homology theory $h_{*}$.

1-Step Stabilization property for CW complex $X$: For every $m$ the image of
the inclusion homomorphism $h_{*}(X^{(m)})\to h_{*}(X)$ stabilizes at
$X^{(m+1)}$.

This property can be written as the formula

$\operatorname{{\rm Ker}}(\phi_{m,m+1})_{*}=\operatorname{{\rm
Ker}}(\phi_{m,\infty})_{*}$

where $\phi_{m,m+\ell}:X^{(m)}\to X^{(m+\ell)}$, $\ell=1,\dots,\infty$,
denotes the inclusions.

We note that every CW complex has the 1-Step Stabilization property for the
ordinary homology $H_{*}$.

By default the 1-Step Stabilization property is assumed for $h_{*}$ in all
dimensions, but often we use this property for fixed $k$ or $k$ depending on
$m$. We denote the set of such $k$ by $K(m)\subset{\mathbb{N}}$. In such
situation we say that $X$ has the 1-Step Stabilization property for $h_{*}$
restricted to $K(m)$.

For example, for the stable homotopy groups $\pi^{s}_{*}$ and, for that matter
for any connective homology theory, the 1-Step Stabilization property
restricted to $K(m)=\\{k\in{\mathbb{N}}\mid k\leq m\\}$ holds for any CW
complex. In this paper we use the 1-Step Stabilization property for the real
$K$-theory, $KO_{*}$, with $k=m,m+1$. Since the complex $K$-theory, $KU_{*}$
is 2-periodic, it is not a restriction on $k$, but in the case of $KO_{*}$ it
is a potential restriction.

######  4.1 Proposition.

The (restricted) 1-Step Stabilization property is a homotopy invariant.

###### Proof.

Suppose that a CW complex $X_{1}$ has the 1-Step Stabilization property for
$h_{*}$. Let $f:X_{1}\to X_{2}$ be a homotopy equivalence with a homotopy
inverse map $g$. We may assume that both maps $f$ and $g$ are cellular. Then
in view of the cellular approximation theorem the composition

$(f|_{X_{1}^{(m+1)}})\circ\phi_{m,m+1}\circ g|_{X_{2}^{(m)}}:X_{2}^{(m)}\to
X_{2}^{(m+1)}$

is homotopic to the inclusion $\phi^{\prime}_{m,m+1}:X_{2}^{(m)}\to
X_{2}^{(m+1)}$. Thus, we have the following commutative diagram

$\begin{CD}h_{k}(X_{1}^{(m)})@>{(\phi_{m,m+1})_{*}}>{}>h_{k}(X_{1}^{(m+1)})@>{(\phi_{m+1,\infty})_{*}}>{}>h_{k}(X_{1})\\\
@A{g|_{*}}A{}A@V{f|_{*}}V{}V@A{g_{*}}A{\cong}A\\\
h_{k}(X_{2}^{(m)})@>{(\phi^{\prime}_{m,m+1})_{*}}>{}>h_{k}(X_{2}^{(m+1)})@>{(\phi^{\prime}_{m+1,\infty})_{*}}>{}>h_{k}(X_{2}).\\\
\end{CD}$

Assume that $(\phi^{\prime}_{m,m+1})_{*}(a)\neq 0$. Then
$(f|)_{*}(\phi_{m,m+1})_{*}(g|)_{*}(a)\neq 0$ and, hence,
$(\phi_{m,m+1})_{*}(g|)_{*}(a)\neq 0$. By the 1-step Stabilization property we
obtain

$0\neq(\phi_{m+1,\infty})_{*}(\phi_{m,m+1})_{*}g|_{*}(a)=g_{*}(\phi^{\prime}_{m+1,\infty})_{*}(f|)_{*}(\phi_{m,m+1})_{*}(g|)_{*}(a)$
$=g_{*}(\phi^{\prime}_{m+1,\infty})_{*}(\phi^{\prime}_{m,m+1})_{*}(a).$

Therefore, $(\phi^{\prime}_{m+1,\infty})_{*}(\phi^{\prime}_{m,m+1})_{*}(a)\neq
0$. ∎

######  4.2 Proposition.

If $X$ has the (restricted) 1-Step Stabilization property for $h_{*}$, then so
does the quotient complex $Y=X/X^{(n)}$.

###### Proof.

We have to check the 1-Step Stabilization property for $Y$ only for $m\geq
n+1$. Let $a\in\operatorname{{\rm Ker}}\\{h_{k}(Y^{(m)})\to h_{k}(Y)\\}$ and
let $\bar{a}\in h_{k}(Y^{(m+1)})$ be the image of $a$. We show that
$\bar{a}=0$. We note that $Y^{(m)}=X^{(m)}/X^{(n)}$. From the diagram
generated by the exact sequence of pairs $(X^{(m)},X^{(n)})$ and $(X,X^{(n)})$
it follows that $a\in\operatorname{{\rm Ker}}\\{\partial:h_{k}(Y^{(m)})\to
h_{k-1}(X^{(n)})\\}$. Hence, by exactness $a$ is the image of some $b\in
h_{k}(X^{(m)})$. If the image of $b$ in $h_{k}(X)$ is trivial, then by the
1-Step Stabilization property its image in $h_{k}(X^{(m+1)})$ is zero and,
therefore, as one can see from the diagram below, $\bar{a}=0$,

$\begin{CD}h_{k}(X^{(n)})@>{}>{}>h_{k}(X^{(n)})@>{}>{}>h_{k}(X^{(n)})\\\
@V{i_{*}}V{}V@V{i_{*}}V{}V@V{i_{*}}V{}V\\\
h_{k}(X^{(m)})@>{}>{}>h_{k}(X^{(m+1)})@>{}>{}>h_{k}(X)\\\
@V{j_{*}}V{}V@V{j_{*}}V{}V@V{j_{*}}V{}V\\\
h_{k}(Y^{(m)})@>{}>{}>h_{k}(Y^{(m+1)})@>{}>{}>h_{k}(Y).\\\ \end{CD}$

Suppose that the image $b^{\prime}\in h_{k}(X)$ of $b$ is not zero. By
exactness, there is $c\in h_{k}(X^{(n)})$ that maps by $i_{*}$ onto
$b^{\prime}$. Then the image of $b-i_{*}(c)$ in $h_{k}(X)$ is zero. By the
1-Step Stabilization, $\bar{b}-i_{*}(c)=0$ where $\bar{b}\in h_{k}(X^{(m+1)})$
is the image of $b$. Note that
$\bar{a}=j_{*}(\bar{b})=j_{*}(\bar{b}-i_{*}(c))=0$. ∎

Since all classifying CW complexes $B\Gamma$ for a group $\Gamma$ are homotopy
equivalent, we can define 1-Step Stabilization property for groups. Thus, a
group $\Gamma$ has the 1-step Stabilization property for a generalized
homology theory $h_{*}$ if a classifying CW complex $B\Gamma$ has it.

We prove the following two propositions

######  4.3 Proposition.

Suppose that a group $\Gamma$ has the 1-Step Stabilization property for
$KO_{*}$ for all $m$, with $k=m$. Then every closed $KO$-inessential manifold
$M$ with the fundamental group $\pi_{1}(M)=\Gamma$ is inessential.

######  4.4 Proposition.

Suppose that an inessential and $KO$-inessential $n$-manifold $M$ has the
fundamental group $\Gamma$, which satisfies the 1-Step Stabilization property
for $KO_{*}$ for all $m$ with $k=m+1$. Then $M$ is strongly inessential.

Proposition 4.3 and Proposition 4.4 lead to the main result of the paper.

######  4.5 Theorem.

Suppose that a group $\Gamma$ has a finite index subgroup $\Gamma^{\prime}$
with the 1-Step Stabilization property for $KO_{*}$ for all $m$ with $k\in
K=\\{m,m+1\\}$ and suppose that $\Gamma^{\prime}$ satisfies the Strong Novikov
Conjecture. Then the original Gromov conjecture holds for all spin
$n$-manifolds, $n>4$, with the fundamental group $\Gamma$.

###### Proof.

Let $M$ be a positive scalar curvature spin $n$-manifold with
$\pi_{1}(M)=\Gamma$. Let $M^{\prime}\to M$ be a finite covering manifold with
the fundamental group $\Gamma^{\prime}$. The Riemannian metric on $M^{\prime}$
lifted from $M$ has a positive scalar curvature. By Rosenberg’s theorem
(Theorem 2.6) $M^{\prime}$ is $KO$-inessential. By Proposition 4.3
$M^{\prime}$ is inessential. By Proposition 4.4 $M^{\prime}$ is strongly
inessential. This completes the proof since $M$ and $M^{\prime}$ have same
universal cover. ∎

Proof of Proposition 4.3

Let $\dim M=n$. By Proposition 4.2 $B\Gamma/B\Gamma^{(n-1)}$ has the 1-Step
Stabilization property property for $KO_{*}$ with $m=n$ and $k=n$. We may
assume that $M$ has one $n$-dimensional cell and
$u(M,M^{(n-1)})\subset(B\Gamma^{(n)},B\Gamma^{(n-1)})$. Since
$u_{*}([M]_{KO})=0$ in $KO_{n}(B\Gamma)$, we obtain $q_{*}u_{*}([M]_{KO})=0$
in $KO_{n}(B\Gamma/B\Gamma^{(n-1)})$ where $q:B\Gamma\to
B\Gamma/B\Gamma^{(n-1)}$ is the quotient map. Let $q^{\prime}$ denote the
restriction of $q$ to $B\Gamma^{(n+1)}$. Since $q^{\prime}_{*}u_{*}([M]_{KO})$
lies in the image of the induced homomorphism for the inclusion

$B\Gamma^{(n)}/B\Gamma^{(n-1)}\to B\Gamma^{(n+1)}/B\Gamma^{(n-1)},$

by the 1-step Stabilization property applied for $m=n$ with $k=n$ we obtain
$q^{\prime}_{*}u_{*}([M]_{KO})=0$ in
$KO_{n}(B\Gamma^{(n+1)}/B\Gamma^{(n-1)})$.

Let $\bar{u}:M/M^{(n-1)}=S^{n}\to B\Gamma^{(n+1)}/B\Gamma^{(n-1)}$ be the
induced map by the classifying map $u$. In the commutative diagram

$\begin{CD}ko_{n}(M)@>{j^{\prime}_{*}}>{}>ko_{n}(S^{n})@>{\bar{u}_{*}}>{}>ko_{n}(B\Gamma^{(n+1)}/B\Gamma^{(n-1)})\\\
@V{per}V{\cong}V@V{\cong}V{per}V\\\
KO_{n}(M)@>{j^{\prime}_{*}}>{}>KO_{n}(S^{n})@>{\bar{u}_{*}}>{}>KO_{n}(B\Gamma^{(n+1)}/B\Gamma^{(n-1)})\\\
\end{CD}$

the $per$ homomorphisms are isomorphisms in view of Proposition 2.3 and
Proposition 2.1. Since $per\circ
j^{\prime}_{*}([M]_{ko})=j^{\prime}_{*}([M]_{KO})$, this implies that
$\bar{u}_{*}j_{*}^{\prime}([M]_{ko})=0$ in
$ko_{n}(B\Gamma^{(n+1)}/B\Gamma^{(n-1)})$. In view of the natural
transformation of homology theories $ko_{*}\to H_{*}(\ ;\mathbb{Z})$ it
follows that $\bar{u}_{*}j_{*}^{\prime}([M])=0$ in
$H_{n}(B\Gamma^{(n+1)}/B\Gamma^{(n-1)})$. Since the homomorphism
$q^{\prime}_{*}:H_{n}(B\Gamma^{(n+1)})\to
H_{n}(B\Gamma^{(n+1)}/B\Gamma^{(n-1)})$ is injective, we obtain that
$u_{*}([M])=0$. Theorem 2.5 completes the proof. ∎

Proof of Proposition 4.4

Let $\dim M=n$. By Proposition 4.2 $B\Gamma$ has the 1-Step Stabilization
property for $KO_{*}$ with $k=m+1$. We may assume that $M$ has a CW complex
structure with one $n$-dimensional cell. Let $\psi:D^{n}\to M$ be its
characteristic map. By Proposition 2.12 we may assume that the classifying map
$u$ satisfies the condition $u(M^{(n-1)})\subset B\Gamma^{(n-2)}$ and
$u(M)\subset B\Gamma^{(n-1)}$.

Note that in the diagram

$\begin{CD}ko_{n}(M/M^{(n-1)})@>{\bar{u}_{*}}>{}>ko_{n}(B\Gamma^{(n-1)}/B\Gamma^{(n-2)})@>{(\bar{\phi}_{n-1,n})_{*}}>{}>ko_{n}(B\Gamma^{(n)}/B\Gamma^{(n-2)})\\\
@V{\cong}V{}V@V{\cong}V{}V@V{\cong}V{}V\\\
KO_{n}(M/M^{(n-1)})@>{\bar{u}_{*}}>{}>KO_{n}(B\Gamma^{(n-1)}/B\Gamma^{(n-2)})@>{(\bar{\phi}_{n-1,n})_{*}}>{}>KO_{n}(B\Gamma^{(n)}/B\Gamma^{(n-2)})\\\
\end{CD}$

the right vertical arrow is an isomorphism in view of Propositions 2.3 and
2.1. The 1-Step Stabilization property of $B\Gamma/B\Gamma^{(n-2)}$ applied
with $m=n-1$ and $k=n$ implies that $(\bar{\phi}_{n-1,n})_{*}\bar{u}_{*}(1)=0$
for $KO_{*}$. The above diagram implies that $i_{*}\bar{u}_{*}(1)=0$ for
$ko_{*}$. Lemma 3.1 completes the proof. ∎

##  5\. Applications to the Gromov Conjecture

For applications of Theorem 4.5 to Gromov’s conjecture we use the following
stronger condition:

$(^{*})_{s}$ The classifying space $B\Gamma$ has the stable homotopy type of
wedge of spheres.

We recall that this means that the spectrum generated by $B\Gamma$ is the
wedge sum of sphere spectra.

We note that in the case when $B\Gamma$ is a finite complex there is $\ell$
such that $\Sigma^{\ell}B\Gamma$ is homotopy equivalent to
$\Sigma^{\ell}(\vee_{j}S^{n_{j}})$ [BCM].

######  5.1 Proposition.

For geometrically finite groups property $(^{*})_{s}$ implies the 1-Step
Stabilization property for any homology theory $h_{*}$.

###### Proof.

Let $f:\Sigma^{\ell}B\Gamma\to\Sigma^{\ell}(\vee S^{n_{j}})$ be a homotopy
equivalence with a homotopy inverse map $g$. We may assume that $f$ and $g$
are cellular maps with respect to the natural CW structure on
$\Sigma^{\ell}S^{n_{j}}=S^{n_{j}+\ell}$. Then the maps
$\Sigma^{\ell}\phi_{m,m+1}$ and $\Sigma^{\ell}\phi_{m,m+1}(g|)(f|)$ are
homotopic where $f|$ and $g|$ are the restrictions of $f$ and $g$ to the
$(m+\ell)$-skeleton. Then for any nontrivial element $a\in
h_{k+\ell}(\Sigma^{\ell}B\Gamma^{(m)})$ with
$(\Sigma^{\ell}\phi_{m,m+1})_{*}(a)\neq 0$ we have $(f|)_{*}(a)\neq 0$. In
this diagram

$\begin{CD}h_{k+\ell}(\Sigma^{\ell}B\Gamma^{(m)})@>{(\Sigma^{\ell}\phi_{m,m+1})_{*}}>{}>h_{k+\ell}(\Sigma^{\ell}B\Gamma^{(m+1)})@>{(\Sigma^{\ell}\phi_{m+1,\infty})_{*}}>{}>h_{k+\ell}(\Sigma^{\ell}B\Gamma)\\\
@V{f|_{*}}V{}V@A{g_{*}}A{\cong}A\\\ h_{k+\ell}(\bigvee_{n_{j}\leq
m}S^{n_{j}+\ell})@>{\xi^{0}_{*}}>{}>h_{k+\ell}(\bigvee_{n_{j}\leq
m+1}S^{n_{j}+\ell})@>{\xi^{1}_{*}}>{}>h_{k+\ell}(\bigvee S^{n_{j}+\ell}).\\\
\end{CD}$

Here the maps $\xi^{0}$ and $\xi^{1}$ are inclusions of a wedge in a large
wedge. Since $\xi^{1}_{*}\xi^{0}_{*}$ is injective, it follows that

$(\phi_{m+1,\infty})_{*}(\phi_{m,m+1})_{*}(a)\neq 0.$

Thus, the 1-Step Stabilization property is verified. ∎

######  5.2 Proposition.

Let $\Gamma_{1}$ and $\Gamma_{2}$ be geometrically finite groups satisfying
the condition $(^{*})_{s}$, then the free product $\Gamma_{1}\ast\Gamma_{2}$
and the product $\Gamma_{1}\times\Gamma_{2}$ satisfy $(^{*})_{s}$.

###### Proof.

Let $\Sigma^{\ell_{1}}B\Gamma_{1}$ and $\Sigma^{\ell_{2}}B\Gamma_{2}$ be
homotopy equivalent to $\vee S^{n_{j}}$ and $\vee S^{m_{i}}$ respectively. We
have $B\left(\Gamma_{1}\ast\Gamma_{2}\right)=B\Gamma_{1}\vee B\Gamma_{2}$, and
thus for any $k$,

$\Sigma^{k}B\left(\Gamma_{1}\ast\Gamma_{2}\right)=\Sigma^{k}B\Gamma_{1}\vee\Sigma^{k}B\Gamma_{2},$

Taking $k=max\\{\ell_{1},\ell_{2}\\},$ we have the result for
$\Gamma_{1}\ast\Gamma_{2}$.

We note that $B\left(\Gamma_{1}\times\Gamma_{2}\right)=B\Gamma_{1}\times
B\Gamma_{2}$. The suspension of product of CW complexes formula [Ha],
$\Sigma(X\times Y)=\Sigma X\vee\Sigma Y\vee\Sigma(X\wedge Y)$ gives us

$\Sigma^{\ell_{1}+\ell_{2}}B\left(\Gamma_{1}\times\Gamma_{2}\right)=\Sigma^{\ell_{1}+\ell_{2}}B\Gamma_{1}\vee\Sigma^{\ell_{1}+\ell_{2}}B\Gamma_{2}\vee\Sigma^{\ell_{1}+\ell_{2}}(B\Gamma_{1}\wedge
B\Gamma_{2}).$

The first two components on the right side are wedge sums of spheres. Since
taking the reduced $k$-suspension is equivalent to the smash product with
$S^{k}$, we obtain for the remaining component

$\Sigma^{\ell_{1}+\ell_{2}}(B\Gamma_{1}\wedge
B\Gamma_{2})=S^{\ell_{1}+\ell_{2}}\wedge(B\Gamma_{1}\wedge
B\Gamma_{2})=(S^{\ell_{1}}\wedge B\Gamma_{1})\wedge(S^{\ell_{2}}\wedge
B\Gamma_{2}).$

Here we used associativity and commutativity of smash product and the fact
that $S^{\ell_{1}+\ell_{2}}=S^{\ell_{1}}\wedge S^{\ell_{2}}$. Thus, the right
hand side is the same as
$(\Sigma^{\ell_{1}}B\Gamma_{1})\wedge(\Sigma^{\ell_{2}}B\Gamma_{2})$, which is
equivalent to

$(\bigvee S^{n_{j}})\wedge(\bigvee S^{m_{i}})=\bigvee(S^{n_{j}}\wedge(\bigvee
S^{m_{i}}))=\bigvee(S^{n_{j}}\wedge S^{m_{i}})=\bigvee S^{n_{j}+m_{i}}.$

∎

######  5.3 Proposition.

The surface groups $\pi_{1}(M_{g})$ for all $g\geq 0$, satisfy the condition
$(^{*})_{s}$.

###### Proof.

The surface $M_{g}$ has one $0$-cell, $2g$ many $1$-cell, one $2$-cell and the
cell structure is obtained by attaching the $2$-cell along
$[a_{1},b_{1}]\cdot[a_{2},b_{2}]\cdots[a_{g},b_{g}]$ to the $1$-skeleton,
which is wedge of $2g$ circles. Let $\varphi:S^{1}\to\bigvee_{2g}S^{1}_{i}$ be
the attaching map for the 2-cell. We claim that
$\Sigma\varphi:S^{2}\to\bigvee_{2g}S^{2}_{i}$ is nullhomotopic. Since
$H_{2}(\bigvee_{2g}S^{2}_{i})=\oplus_{i=1}^{2g}H_{2}(S^{2}_{i})$, by Hurewicz
Theorem $\pi_{2}(\bigvee_{2g}S^{2}_{i})=\oplus_{i=1}^{2g}\pi_{2}(S^{2}_{i})$.
Thus,

$[\Sigma\varphi]=([\Sigma(p_{1}\varphi)],\dots,[\Sigma(p_{2g}\varphi])\in\bigoplus_{i=1}^{2g}\pi_{2}(S^{2}_{i})$

where $p_{i}:\bigvee_{2g}S^{1}\to S^{1}_{i}$ is the projection to $i$-th
summand. Note that each $p_{i}\circ\varphi:S^{1}\to S^{1}$ has degree 0. Hence
$deg\Sigma(p_{i}\varphi)=deg(p_{i}\varphi)=0$. Hence $[\Sigma\varphi]=0$ and
$\Sigma\varphi$ is null-homotopic. Therefore, $\Sigma M_{g}$ is homotopy
equivalent to $\left(\bigvee_{2g}S_{i}^{2}\right)\vee S^{3}$. ∎

Let $X$ be a finite simplicial graph with the set of vertices
$S=\\{s_{1},s_{2},...,s_{n}\\}$, and the set of edges $E$. We recall, that the
right-angled Artin group defined by $X$ is the group with following
presentation

$A_{X}=\langle S\mid[s_{i},s_{j}]=1\ \ \text{whenever}\ \ \\{s_{i},s_{j}\\}\in
E\rangle.$

The Salvetti complex, $\mathcal{S}_{X}$, which is a classifying space,
$\mathcal{S}_{X}:=BA_{X}$ for the group $A_{X}$ is constructed as follows:
Begin with a wedge of circles attached to a point $x_{0}$ and labeled by the
generators $s_{1},\cdots,s_{n}$. For each edge, $\\{s_{i},s_{j}\\}\in E$,
consider the $2$-torus obtained by attaching $2$-cell to the wedge of circles
corresponding to $s_{i}$ and $s_{j}$ along the relation $[s_{i},s_{j}]$. For
each triangle in $X$ connecting three vertices $s_{i},s_{j},s_{k}$, consider
the $3$-torus defined by circles corresponding to them. Continue this process,
having a $k$-torus for each set of $k$ mutually commuting generators (i.e.,
generators spanning a complete subgraph in $X$). The resulting space is the
Salvetti complex $\mathcal{S}_{X}$ for $A_{X}$.

######  5.4 Proposition.

Suppose that $K=e^{0}\cup(\cup_{i\in J}e^{n_{i}}_{i})$ is a subcomplex of an
$n$-dimensional torus $T^{n}=S^{1}\times\cdots\times S^{1}$ with the product
CW complex structure and $S^{1}=e^{0}\cup e^{1}$. Then for sufficiently large
$\ell$ the reduced $\ell$-suspension $\Sigma^{\ell}K$ is homotopy equivalent
to the wedge of spheres $\vee_{i\in J}S^{n_{i}+\ell}$.

Moreover, there is a cellular homotopy equivalence
$f:\Sigma^{\ell}K\to\vee_{i\in J}S^{n_{i}+\ell}$ which takes the cell
$\Sigma^{\ell}e^{n_{i}}_{i}$ to $S^{n_{i}+\ell}$ for all $i\in J$.

###### Proof.

The proof is based on the formula

$\Sigma(X\times Y)=\Sigma X\vee\Sigma Y\vee\Sigma(X\wedge Y).$

We refer to the proof of Proposition 6.5 in [Dr2] for details. ∎

######  5.5 Corollary.

Any subcomplex of $n$-dimensional torus $T^{n}$ satisfies $(^{*})_{s}$.

######  5.6 Proposition.

RAAGs satisfy $(^{*})_{s}$.

###### Proof.

Let $A_{X}$ be right-angled Artin group generated by $n$ generators where $X$
is the corresponding simple graph with $n$ vertices
$\\{s_{1},\cdots,s_{n}\\}$. Since the Salvetti complex $\mathcal{S}_{X}$ is
built as a union of $k$-tori, $T^{k}\subset T^{n}$ for each complete
$k$-subgraph of $X$, clearly $\mathcal{S}_{X}$ is a subcomplex of $T^{n}$.
Thus Lemma 5.5, completes the proof. ∎

In view of Proposition 5.1, the condition $(^{*})_{s}$ implies the 1-Step
Stabilization property for $KO_{*}$. Thus, Theorem 4.5 implies the following

######  5.7 Corollary.

Suppose that a group $\Gamma$ satisfies $(^{*})_{s}$ and the Strong Novikov
conjecture holds true for $\Gamma$. Then the Strong Gromov conjecture holds
for spin manifolds with the fundamental group $\Gamma$.

######  5.8 Corollary.

The Strong Gromov Conjecture holds true for spin manifolds with the
fundamental group $\Gamma$ where

  * •

$\Gamma$ is a RAAG,

  * •

$\Gamma$ is a finite product of orientable surface groups.

###### Proof.

Since these groups are CAT(0) groups, the Strong Novikov Conjecture holds for
them and the result follows. ∎

In view of Proposition 4.5, if the Strong Gromov Conjecture holds for a group
$\Gamma^{\prime}$ contained in $\Gamma$ as a finite index subgroup, then the
original Gromov’s conjecture holds for $\Gamma$. Since the fundamental group
of non-orientable surface contains as index 2, the fundamental group of an
orientable surface, we obtain that the original Gromov Conjecture holds true
for products of non-orientable surface groups.

######  5.9 Remark.

It is well known that each RAAG is contained in some right-angled Coxeter
groups as finite index subgroup, [DJ], hence the Gromov Conjecture holds true
for spin manifolds having fundamental groups those right-angled Coxeter
groups.

######  5.10 Question.

Does the property $(^{*})_{s}$ implies the (Strong) Novikov conjecture for a
group $\Gamma$?

###  5.1. More groups satisfying 1-Step Stabilization Property

Given an abelian group $G$ and $n\geq 1$, a CW complex $X$ having trivial
reduced homology $\tilde{H}_{i}(X)=0$ for $i\neq n$ and $H_{n}(X)=G$ is called
a Moore space and is denoted as $M(G,n)$. In the case of a finitely generated
group $G$ and $n\geq 2$ the Moore space can be chosen to be the finite wedge
of $n$-sphere $S^{n}$ and $n$-spheres with one $(n+1)$-dimensional cell
attached $S^{n}\cup_{\phi}D^{n+1}$.

The condition $(^{*})_{s}$ can be weakened to the following

$(^{*})_{m}$ The classifying space $B\Gamma$ has the stable homotopy type of a
wedge of Moore spaces.

######  5.11 Proposition.

For geometrically finite groups property $(^{*})_{m}$ implies the 1-Step
Stabilization property for any homology theory $h_{*}$.

###### Proof.

Let $f:\Sigma^{\ell}B\Gamma\to\Sigma^{\ell}(\vee M(G_{j},n_{j}))$ be a
homotopy equivalence with a homotopy inverse map $g$. Since $B\Gamma$ is
geometrically finite, the groups $G_{j}$ are finitely generated. We may assume
that each $G_{j}$ is isomorphic to ${\mathbb{Z}}$ or to ${\mathbb{Z}}_{m_{j}}$
with $m_{j}=p^{k_{j}}$ for a prime number $p$. Thus, $M(G_{j},n_{j})$ is
either an $n_{j}$-sphere $S^{n_{j}}$ or an $n_{j}$-sphere with
$(n_{j}+1)$-cell attached, $S^{n_{j}}\cup_{\psi_{j}}D^{n_{j}+1}$. We may
assume that $f$ and $g$ are cellular map with respect to the natural CW
structure on $\Sigma^{\ell}M(G_{j},n_{j})=M(G_{j},n_{j}+\ell)$. Let
$m^{\prime}=m+\ell$, $k^{\prime}=k+\ell$, and $n_{j}^{\prime}=n_{j}+\ell$. Let
$f^{m}$ and $g^{m}$ denote the restrictions of $f$ and $g$ to the
$(m+\ell)$-skeletons. Then for any nontrivial element $a\in
h_{k+\ell}(\Sigma^{\ell}B\Gamma^{(m)})$ with
$(\Sigma^{\ell}\phi_{m,m+1})_{*}(a)\neq 0$ we have $(f|)_{*}(a)\neq 0$:

Let $\phi^{0}=\Sigma^{\ell}\phi_{m,m+1}$ and
$\phi^{1}=\Sigma^{\ell}\phi_{m+1,\infty}$. Consider the diagram

$\begin{CD}h_{k^{\prime}}(\Sigma^{\ell}B\Gamma^{(m)})@>{\phi^{0}_{*}}>{}>h_{k^{\prime}}(\Sigma^{\ell}B\Gamma^{(m+1)})@>{\phi^{1}_{*}}>{}>h_{k^{\prime}}(\Sigma^{\ell}B\Gamma)\\\
@A{g^{m}_{*}}A{}A@V{f^{m+1}_{*}}V{}V@A{g_{*}}A{\cong}A\\\
h_{k^{\prime}}(\bigvee
M(G_{j},n_{j}^{\prime})^{(m^{\prime})})@>{\xi^{0}_{*}}>{}>h_{k^{\prime}}(\bigvee
M(G_{j},n_{j}^{\prime})^{(m^{\prime}+1)})@>{\xi^{1}_{*}}>{}>h_{k^{\prime}}(\bigvee
M(G_{j},n_{j}^{\prime})).\\\ \end{CD}$

where $\xi^{0}$ and $\xi^{1}$ are the inclusions of wedge sums. Let $a\in
h_{k^{\prime}}(\Sigma^{\ell}B\Gamma^{(m)})$ with $\phi^{0}_{*}(a)\neq 0$. We
need to show that $\phi^{1}_{*}\phi^{0}_{*}(a)\neq 0$.

Since the maps $\phi^{0}$ and $\phi^{0}g^{m}f^{m}$ are homotopic, we obtain

$g^{m+1}_{*}\xi^{0}(f^{m}_{*}(a))=\phi^{0}_{*}g^{m}_{*}(f^{m}_{*}(a))=\phi^{0}_{*}(a)\neq
0.$

Therefore, $\xi^{0}_{*}(f^{m}_{*}(a))\neq 0$. We note that

$\bigvee M(G_{j},n_{j}^{\prime})^{(m^{\prime}+1)}=\bigvee_{n_{j}\leq
m^{\prime}}M(G_{j},n_{j}^{\prime})\vee(\bigvee S^{m^{\prime}+1})$

and $\xi^{0}_{*}(f^{m}_{*}(a))$ lives in the summand $\bigvee_{n_{j}\leq
m^{\prime}}M(G_{j},n_{j}^{\prime})$ which is a retract of $\bigvee
M(G_{j},n_{j}^{\prime})$. Hence, $\xi^{1}_{*}(\xi^{0}(f^{m}_{*}(a))\neq 0$.
Then

$\phi^{1}_{*}\phi^{0}_{*}(a)=\phi^{1}_{*}g^{m+1}_{*}\xi^{0}_{*}f^{m}_{*}(a)=g_{*}\xi^{1}_{*}\xi^{0}_{*}f^{m}_{*}(a)\neq
0.$

Thus, the 1-Step Stabilization property is verified. ∎

######  5.12 Proposition.

All finitely generated groups with $\dim B\Gamma\leq 2$ have property
$(^{*})_{m}$ with a finite wedge of Moore spaces.

###### Proof.

Apply Proposition 2.3. ∎

Suppose that a geometrically finite group $\Gamma$ has property $(^{*})_{m}$.
Then for some $\ell$ the $\ell$-suspension $\Sigma^{\ell}B\Gamma$ is homotopy
equivalent to a wedge $\vee_{j=1}^{s}M(G_{j},n_{j})$ where
$G_{j}={\mathbb{Z}}$ or ${\mathbb{Z}}_{p^{k_{j}}}$ where $p$ is a prime number
and $n_{j}>\ell$. We call such $\Gamma$ 2-avoiding if $p^{k_{j}}\neq 2$ for
all $j$.

######  5.13 Proposition.

( [JN] Corollary 6.6) If $gcd(k,l)=d$ is odd or $4$ divides $k$, then
$M({\mathbb{Z}}_{k},m)\wedge M({\mathbb{Z}}_{l},n)$ is homotopy equivalent to
$M({\mathbb{Z}}_{d},m+n)\vee M({\mathbb{Z}}_{d},m+n+1)$.

######  5.14 Proposition.

The product of geometrically finite groups $\Gamma_{1}\times\Gamma_{2}$ has
property $(^{*})_{m}$, when each of $\Gamma_{1}$, $\Gamma_{2}$ has property
$(^{*})_{m}$, provided one of the groups is 2-avoiding.

###### Proof.

Proceeding exactly similar to the proof of Proposition 5.2, this time
$(\Sigma^{\ell_{1}}B\Gamma_{1})\wedge(\Sigma^{\ell_{2}}B\Gamma_{2})$ is
homotopy equivalent to

$\left(\vee_{j=1}^{s}M(G_{j},n_{j})\right)\bigwedge\left(\vee_{i=1}^{t}M(G_{i},m_{i})\right)=\bigvee_{i,j}M(G_{j},n_{j})\wedge
M(G_{i},m_{i}).$

Since $M({\mathbb{Z}},n)=S^{n}$, if any of $G_{j}$ or $G_{i}$ is
${\mathbb{Z}}$ we have $M(G_{j},n_{j})\wedge M(G_{i},m_{i})$ to be a
suspension of one of the Moore spaces, which is again Moore space. It is
enough to consider when we have same prime $p$ for both groups $G_{j}$ and
$G_{i}$, since for $k=rs$ with $r,s$ relatively prime we have
$M({\mathbb{Z}}_{k},n)=M({\mathbb{Z}}_{r},n)\vee M({\mathbb{Z}}_{s},n)$. If
$G_{j}={\mathbb{Z}}_{p^{j}}$ and $G_{i}={\mathbb{Z}}_{p^{i}}$, where $p$ is
some prime, then assuming $i\leq j$, we have $gcd(p^{j},p^{i})=p^{i}$, thus by
Proposition 5.13 we have,

$M({\mathbb{Z}}_{p^{j}},n_{j})\wedge
M({\mathbb{Z}}_{p^{i}},m_{i})=M({\mathbb{Z}}_{p^{i}},n_{j}+m_{i})\vee
M({\mathbb{Z}}_{p^{i}},n_{j}+m_{i}+1).$

Note that if $p=2$, $1\leq i$ and $2\ \leq j$ then Proposition 5.13 gives the
above splitting. ∎

######  5.15 Corollary.

The Strong Gromov Conjecture holds for spin manifolds whose fundamental groups
are finite products $\Gamma=\Gamma_{1}\times\dots\times\Gamma_{k}$ of
2-dimensional finitely generated 2-avoiding groups such that each $\Gamma_{i}$
satisfies the Strong Novikov Conjecture.

We note that 2-avoiding torsion free one-relator groups are among examples of
above $\Gamma_{i}$. Since they have finite asymptotic dimension [BD], [Dr3],
[T], they satisfy the coarse Baum-Connes conjecture for both $KU$ and $KO$.
Therefore, they satisfy the Strong Gromov Conjecture [HR]. By the Lyndon-
Cockeroft theorem [L],[C] they are 2-dimensional.

We recall that a 2-dimensional group is 2-avoiding if its abelianization does
not contain $\mathbb{Z}_{2}$ as a direct summand. An affirmative answer to the
following question would prove the original Gromov’s conjecture for the
products of all 2-dimensional groups.

######  5.16 Question.

Does every 2-dimensional group contain a 2-avoiding finite index subgroup?

We believe that the other required condition for 2-dimensional groups in
Corollary 5.15, the Strong Novikov Conjecture, can be derived from results of
Mathai [Mat] and Hanke-Schick [HS].

###  5.2. Bolotov’s question.

Dmitri Bolotov constructed an example $M_{b}$ of a closed inessential spin
4-manifold which is not strongly inessential. At the end of his paper he asked
if any product with a torus $M_{b}\times T^{p}$ can carry a metric of positive
scalar curvature [Bo2]. Here we give a negative answer.

######  5.17 Proposition.

The manifolds $M_{b}\times T^{p}$ for all $p>0$ do not admit a metric with
positive scalar curvature.

###### Proof.

The fundamental group of Bolotov’s manifold is
$\Gamma=\mathbb{Z}\ast\mathbb{Z}^{3}$. By Proposition 5.2 the group
$\Gamma\times{\mathbb{Z}}^{p}$ satisfies the condition $(^{*})_{s}$. The
Strong Novikov conjecture holds for the group $\Gamma\times{\mathbb{Z}}^{p}$
since it is constructed out of integers by taking operation of the product and
the free product. Note that $M_{b}\times T^{p}$ is spin and inessential. By
Theorem 4.5 the Strong Gromov conjecture holds for $M_{b}\times T^{p}$ for
$p>0$. Therefore, $M_{b}\times T^{p}$ cannot carry a metric of positive scalar
curvature. Hence $M_{b}$ cannot carry a metric of positive scalar curvature as
well. ∎

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